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Techniques of Integration

- Calculus
- Absolute Maxima and Minima
- Absolute and Conditional Convergence
- Accumulation Function
- Accumulation Problems
- Algebraic Functions
- Alternating Series
- Antiderivatives
- Application of Derivatives
- Approximating Areas
- Arc Length of a Curve
- Arithmetic Series
- Average Value of a Function
- Calculus of Parametric Curves
- Candidate Test
- Combining Differentiation Rules
- Combining Functions
- Continuity
- Continuity Over an Interval
- Convergence Tests
- Cost and Revenue
- Density and Center of Mass
- Derivative Functions
- Derivative of Exponential Function
- Derivative of Inverse Function
- Derivative of Logarithmic Functions
- Derivative of Trigonometric Functions
- Derivatives
- Derivatives and Continuity
- Derivatives and the Shape of a Graph
- Derivatives of Inverse Trigonometric Functions
- Derivatives of Polar Functions
- Derivatives of Sec, Csc and Cot
- Derivatives of Sin, Cos and Tan
- Determining Volumes by Slicing
- Direction Fields
- Disk Method
- Divergence Test
- Eliminating the Parameter
- Euler's Method
- Evaluating a Definite Integral
- Evaluation Theorem
- Exponential Functions
- Finding Limits
- Finding Limits of Specific Functions
- First Derivative Test
- Function Transformations
- General Solution of Differential Equation
- Geometric Series
- Growth Rate of Functions
- Higher-Order Derivatives
- Hydrostatic Pressure
- Hyperbolic Functions
- Implicit Differentiation Tangent Line
- Implicit Relations
- Improper Integrals
- Indefinite Integral
- Indeterminate Forms
- Initial Value Problem Differential Equations
- Integral Test
- Integrals of Exponential Functions
- Integrals of Motion
- Integrating Even and Odd Functions
- Integration Formula
- Integration Tables
- Integration Using Long Division
- Integration of Logarithmic Functions
- Integration using Inverse Trigonometric Functions
- Intermediate Value Theorem
- Inverse Trigonometric Functions
- Jump Discontinuity
- Lagrange Error Bound
- Limit Laws
- Limit of Vector Valued Function
- Limit of a Sequence
- Limits
- Limits at Infinity
- Limits of a Function
- Linear Approximations and Differentials
- Linear Differential Equation
- Linear Functions
- Logarithmic Differentiation
- Logarithmic Functions
- Logistic Differential Equation
- Maclaurin Series
- Manipulating Functions
- Maxima and Minima
- Maxima and Minima Problems
- Mean Value Theorem for Integrals
- Models for Population Growth
- Motion Along a Line
- Motion in Space
- Natural Logarithmic Function
- Net Change Theorem
- Newton's Method
- Nonhomogeneous Differential Equation
- One-Sided Limits
- Optimization Problems
- P Series
- Particle Model Motion
- Particular Solutions to Differential Equations
- Polar Coordinates
- Polar Coordinates Functions
- Polar Curves
- Population Change
- Power Series
- Ratio Test
- Removable Discontinuity
- Riemann Sum
- Rolle's Theorem
- Root Test
- Second Derivative Test
- Separable Equations
- Simpson's Rule
- Solid of Revolution
- Solutions to Differential Equations
- Surface Area of Revolution
- Symmetry of Functions
- Tangent Lines
- Taylor Polynomials
- Taylor Series
- Techniques of Integration
- The Fundamental Theorem of Calculus
- The Mean Value Theorem
- The Power Rule
- The Squeeze Theorem
- The Trapezoidal Rule
- Theorems of Continuity
- Trigonometric Substitution
- Vector Valued Function
- Vectors in Calculus
- Vectors in Space
- Washer Method
- Decision Maths
- Geometry
- 2 Dimensional Figures
- 3 Dimensional Vectors
- 3-Dimensional Figures
- Altitude
- Angles in Circles
- Arc Measures
- Area and Volume
- Area of Circles
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- Area of Parallelograms
- Area of Plane Figures
- Area of Rectangles
- Area of Regular Polygons
- Area of Rhombus
- Area of Trapezoid
- Area of a Kite
- Composition
- Congruence Transformations
- Congruent Triangles
- Convexity in Polygons
- Coordinate Systems
- Dilations
- Distance and Midpoints
- Equation of Circles
- Equilateral Triangles
- Figures
- Fundamentals of Geometry
- Geometric Inequalities
- Geometric Mean
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- Glide Reflections
- HL ASA and AAS
- Identity Map
- Inscribed Angles
- Isometry
- Isosceles Triangles
- Law of Cosines
- Law of Sines
- Linear Measure and Precision
- Median
- Parallel Lines Theorem
- Parallelograms
- Perpendicular Bisector
- Plane Geometry
- Polygons
- Projections
- Properties of Chords
- Proportionality Theorems
- Pythagoras Theorem
- Rectangle
- Reflection in Geometry
- Regular Polygon
- Rhombuses
- Right Triangles
- Rotations
- SSS and SAS
- Segment Length
- Similarity
- Similarity Transformations
- Special quadrilaterals
- Squares
- Surface Area of Cone
- Surface Area of Cylinder
- Surface Area of Prism
- Surface Area of Sphere
- Surface Area of a Solid
- Surface of Pyramids
- Symmetry
- Translations
- Trapezoids
- Triangle Inequalities
- Triangles
- Using Similar Polygons
- Vector Addition
- Vector Product
- Volume of Cone
- Volume of Cylinder
- Volume of Pyramid
- Volume of Solid
- Volume of Sphere
- Volume of prisms
- Mechanics Maths
- Acceleration and Time
- Acceleration and Velocity
- Angular Speed
- Assumptions
- Calculus Kinematics
- Coefficient of Friction
- Connected Particles
- Constant Acceleration
- Constant Acceleration Equations
- Converting Units
- Force as a Vector
- Kinematics
- Newton's First Law
- Newton's Law of Gravitation
- Newton's Second Law
- Newton's Third Law
- Projectiles
- Pulleys
- Resolving Forces
- Statics and Dynamics
- Tension in Strings
- Variable Acceleration
- Probability and Statistics
- Bar Graphs
- Basic Probability
- Charts and Diagrams
- Conditional Probabilities
- Continuous and Discrete Data
- Frequency, Frequency Tables and Levels of Measurement
- Independent Events Probability
- Line Graphs
- Mean Median and Mode
- Mutually Exclusive Probabilities
- Probability Rules
- Probability of Combined Events
- Quartiles and Interquartile Range
- Systematic Listing
- Pure Maths
- ASA Theorem
- Absolute Value Equations and Inequalities
- Addition and Subtraction of Rational Expressions
- Addition, Subtraction, Multiplication and Division
- Algebra
- Algebraic Fractions
- Algebraic Notation
- Algebraic Representation
- Analyzing Graphs of Polynomials
- Angle Measure
- Angles
- Angles in Polygons
- Approximation and Estimation
- Area and Circumference of a Circle
- Area and Perimeter of Quadrilaterals
- Area of Triangles
- Arithmetic Sequences
- Average Rate of Change
- Bijective Functions
- Binomial Expansion
- Binomial Theorem
- Chain Rule
- Circle Theorems
- Circles
- Circles Maths
- Combination of Functions
- Combinatorics
- Common Factors
- Common Multiples
- Completing the Square
- Completing the Squares
- Complex Numbers
- Composite Functions
- Composition of Functions
- Compound Interest
- Compound Units
- Conic Sections
- Construction and Loci
- Converting Metrics
- Convexity and Concavity
- Coordinate Geometry
- Coordinates in Four Quadrants
- Cubic Function Graph
- Cubic Polynomial Graphs
- Data transformations
- Deductive Reasoning
- Definite Integrals
- Deriving Equations
- Determinant of Inverse Matrix
- Determinants
- Differential Equations
- Differentiation
- Differentiation Rules
- Differentiation from First Principles
- Differentiation of Hyperbolic Functions
- Direct and Inverse proportions
- Disjoint and Overlapping Events
- Disproof by Counterexample
- Distance from a Point to a Line
- Divisibility Tests
- Double Angle and Half Angle Formulas
- Drawing Conclusions from Examples
- Ellipse
- Equation of Line in 3D
- Equation of a Perpendicular Bisector
- Equation of a circle
- Equations
- Equations and Identities
- Equations and Inequalities
- Estimation in Real Life
- Euclidean Algorithm
- Evaluating and Graphing Polynomials
- Even Functions
- Exponential Form of Complex Numbers
- Exponential Rules
- Exponentials and Logarithms
- Expression Math
- Expressions and Formulas
- Faces Edges and Vertices
- Factorials
- Factoring Polynomials
- Factoring Quadratic Equations
- Factorising expressions
- Factors
- Finding Maxima and Minima Using Derivatives
- Finding Rational Zeros
- Finding the Area
- Forms of Quadratic Functions
- Fractional Powers
- Fractional Ratio
- Fractions
- Fractions and Decimals
- Fractions and Factors
- Fractions in Expressions and Equations
- Fractions, Decimals and Percentages
- Function Basics
- Functional Analysis
- Functions
- Fundamental Counting Principle
- Fundamental Theorem of Algebra
- Generating Terms of a Sequence
- Geometric Sequence
- Gradient and Intercept
- Graphical Representation
- Graphing Rational Functions
- Graphing Trigonometric Functions
- Graphs
- Graphs and Differentiation
- Graphs of Common Functions
- Graphs of Exponents and Logarithms
- Graphs of Trigonometric Functions
- Greatest Common Divisor
- Growth and Decay
- Growth of Functions
- Highest Common Factor
- Hyperbolas
- Imaginary Unit and Polar Bijection
- Implicit differentiation
- Inductive Reasoning
- Inequalities Maths
- Infinite geometric series
- Injective functions
- Instantaneous Rate of Change
- Integers
- Integrating Polynomials
- Integrating Trig Functions
- Integrating e^x and 1/x
- Integration
- Integration Using Partial Fractions
- Integration by Parts
- Integration by Substitution
- Integration of Hyperbolic Functions
- Interest
- Inverse Hyperbolic Functions
- Inverse Matrices
- Inverse and Joint Variation
- Inverse functions
- Iterative Methods
- Law of Cosines in Algebra
- Law of Sines in Algebra
- Laws of Logs
- Limits of Accuracy
- Linear Expressions
- Linear Systems
- Linear Transformations of Matrices
- Location of Roots
- Logarithm Base
- Logic
- Lower and Upper Bounds
- Lowest Common Denominator
- Lowest Common Multiple
- Math formula
- Matrices
- Matrix Addition and Subtraction
- Matrix Determinant
- Matrix Multiplication
- Metric and Imperial Units
- Misleading Graphs
- Mixed Expressions
- Modulus Functions
- Modulus and Phase
- Multiples of Pi
- Multiplication and Division of Fractions
- Multiplicative Relationship
- Multiplying and Dividing Rational Expressions
- Natural Logarithm
- Natural Numbers
- Notation
- Number
- Number Line
- Number Systems
- Numerical Methods
- Odd functions
- Open Sentences and Identities
- Operation with Complex Numbers
- Operations with Decimals
- Operations with Matrices
- Operations with Polynomials
- Order of Operations
- Parabola
- Parallel Lines
- Parametric Differentiation
- Parametric Equations
- Parametric Integration
- Partial Fractions
- Pascal's Triangle
- Percentage
- Percentage Increase and Decrease
- Percentage as fraction or decimals
- Perimeter of a Triangle
- Permutations and Combinations
- Perpendicular Lines
- Points Lines and Planes
- Polynomial Graphs
- Polynomials
- Powers Roots And Radicals
- Powers and Exponents
- Powers and Roots
- Prime Factorization
- Prime Numbers
- Problem-solving Models and Strategies
- Product Rule
- Proof
- Proof and Mathematical Induction
- Proof by Contradiction
- Proof by Deduction
- Proof by Exhaustion
- Proof by Induction
- Properties of Exponents
- Proportion
- Proving an Identity
- Pythagorean Identities
- Quadratic Equations
- Quadratic Function Graphs
- Quadratic Graphs
- Quadratic functions
- Quadrilaterals
- Quotient Rule
- Radians
- Radical Functions
- Rates of Change
- Ratio
- Ratio Fractions
- Rational Exponents
- Rational Expressions
- Rational Functions
- Rational Numbers and Fractions
- Ratios as Fractions
- Real Numbers
- Reciprocal Graphs
- Recurrence Relation
- Recursion and Special Sequences
- Remainder and Factor Theorems
- Representation of Complex Numbers
- Rewriting Formulas and Equations
- Roots of Complex Numbers
- Roots of Polynomials
- Roots of Unity
- Rounding
- SAS Theorem
- SSS Theorem
- Scalar Triple Product
- Scale Drawings and Maps
- Scale Factors
- Scientific Notation
- Second Order Recurrence Relation
- Sector of a Circle
- Segment of a Circle
- Sequences
- Sequences and Series
- Series Maths
- Sets Math
- Similar Triangles
- Similar and Congruent Shapes
- Simple Interest
- Simplifying Fractions
- Simplifying Radicals
- Simultaneous Equations
- Sine and Cosine Rules
- Small Angle Approximation
- Solving Linear Equations
- Solving Linear Systems
- Solving Quadratic Equations
- Solving Radical Inequalities
- Solving Rational Equations
- Solving Simultaneous Equations Using Matrices
- Solving Systems of Inequalities
- Solving Trigonometric Equations
- Solving and Graphing Quadratic Equations
- Solving and Graphing Quadratic Inequalities
- Special Products
- Standard Form
- Standard Integrals
- Standard Unit
- Straight Line Graphs
- Substraction and addition of fractions
- Sum and Difference of Angles Formulas
- Sum of Natural Numbers
- Surds
- Surjective functions
- Tables and Graphs
- Tangent of a Circle
- The Quadratic Formula and the Discriminant
- Transformations
- Transformations of Graphs
- Translations of Trigonometric Functions
- Triangle Rules
- Triangle trigonometry
- Trigonometric Functions
- Trigonometric Functions of General Angles
- Trigonometric Identities
- Trigonometric Ratios
- Trigonometry
- Turning Points
- Types of Functions
- Types of Numbers
- Types of Triangles
- Unit Circle
- Units
- Variables in Algebra
- Vectors
- Verifying Trigonometric Identities
- Writing Equations
- Writing Linear Equations
- Statistics
- Bias in Experiments
- Binomial Distribution
- Binomial Hypothesis Test
- Bivariate Data
- Box Plots
- Categorical Data
- Categorical Variables
- Central Limit Theorem
- Chi Square Test for Goodness of Fit
- Chi Square Test for Homogeneity
- Chi Square Test for Independence
- Chi-Square Distribution
- Combining Random Variables
- Comparing Data
- Comparing Two Means Hypothesis Testing
- Conditional Probability
- Conducting a Study
- Conducting a Survey
- Conducting an Experiment
- Confidence Interval for Population Mean
- Confidence Interval for Population Proportion
- Confidence Interval for Slope of Regression Line
- Confidence Interval for the Difference of Two Means
- Confidence Intervals
- Correlation Math
- Cumulative Frequency
- Data Analysis
- Data Interpretation
- Discrete Random Variable
- Distributions
- Dot Plot
- Empirical Rule
- Errors in Hypothesis Testing
- Estimator Bias
- Events (Probability)
- Frequency Polygons
- Generalization and Conclusions
- Geometric Distribution
- Histograms
- Hypothesis Test for Correlation
- Hypothesis Test of Two Population Proportions
- Hypothesis Testing
- Inference for Distributions of Categorical Data
- Inferences in Statistics
- Large Data Set
- Least Squares Linear Regression
- Linear Interpolation
- Linear Regression
- Measures of Central Tendency
- Methods of Data Collection
- Normal Distribution
- Normal Distribution Hypothesis Test
- Normal Distribution Percentile
- Point Estimation
- Probability
- Probability Calculations
- Probability Distribution
- Probability Generating Function
- Quantitative Variables
- Quartiles
- Random Variables
- Randomized Block Design
- Residual Sum of Squares
- Residuals
- Sample Mean
- Sample Proportion
- Sampling
- Sampling Distribution
- Scatter Graphs
- Single Variable Data
- Skewness
- Standard Deviation
- Standard Normal Distribution
- Statistical Graphs
- Statistical Measures
- Stem and Leaf Graph
- Sum of Independent Random Variables
- Survey Bias
- Transforming Random Variables
- Tree Diagram
- Two Categorical Variables
- Two Quantitative Variables
- Type I Error
- Type II Error
- Types of Data in Statistics
- Venn Diagrams

Before getting into the math, here is a joke to lighten the mood (who said math couldn't be funny?):

Teacher: What is 3+3?

Bobby: 3!

Teacher: Yes, Bobby, that is correct.

Jokes aside, you have probably come across factorials in a variety of situations. Normally, we define factorials by writing:

$$n! = n(n-1)...1$$

This definition makes sense if n is a positive integer, but not if \(n\) is any other type of real number. Mathematicians, being mathematicians, decided that this state of things was, frankly, unacceptable. So, they came up with definitions of factorials that allow you to do things like find \(\pi!\). One of these is the gamma function

$$n! = \Gamma(n+1) = \int_0^{\infty} x^{n-1}e^{-x} \; dx.$$

This is great, but presents us with another dilemma: how do we evaluate this (frankly somewhat scary) integral? Fortunately, there are a variety of techniques of integration we can throw at the problem. The most common techniques of integration are:

The Power Rule for Integration

Integration by Substitution

Integration by Parts

Integration by Partial Fractions

Integrating Functions Using Long Division

These are essential to know, but won't always help you with integrals like the gamma function. There are too many integration techniques for any one article to cover fully, but here you will see a few of the better-known techniques for dealing with problematic integrals. In particular, this article will cover the power rule for integration, integrals of inverse functions, Weierstrass substitution, and Feynman's technique of integration. For more details on the other techniques of integration listed above, see the corresponding articles.

Just like there is a power rule for differentiation, there is a power rule for integration.

The **Power Rule for Integration** states that

$$\displaystyle\int ax^n \; dx = \frac{a}{n+1}x^{n+1} + C.$$

Here, \(n\) can be any real number (positive, negative, zero, integer, rational, or irrational).

We can prove the power rule for integration directly from the power rule for differentiation. Recall that the power rule for differentiation states that

$$\frac{d}{dx}x^n = nx^{n-1}.$$

Given a function

$$\frac{a}{n+1}x^{n+1},$$

using the power rule for differentiation, we know that

$$\begin{align}\frac{d}{dx}\frac{a}{n+1}x^{n+1} &= \frac{a}{n+1}\frac{d}{dx} x^{n+1}\\ &= \frac{a}{n+1}\left((n+1)x^n\right)\\ &= ax^n.\end{align}$$

Integrating both sides of the equation and using the Fundamental Theorem of Calculus, we get that

\[\int \frac{d}{dx}\frac{a}{n+1}x^{n+1}\;dx = \int ax^n \; dx. \]

or in other words

\[ \frac{a}{n+1}x^{n+1} + C = \int ax^n \; dx.\]

You can think of the power rule for integration as 'undoing' the power rule for differentiation. Many other common integration techniques are based on this strategy of 'undoing' differentiation rules.

Evaluate the integral

\[\int 3x^3 \; dx.\]

**Solution:**

Here, you can use the Power Rule for Integration with \(a=3\) and \(n=3\). So,

\[\int 3x^3 \; dx = \frac{3}{4}x^{4} + C.\]

You can check by differentiating that this answer is correct.

Some common techniques of integration include:

Integration by Substitution

Integration by Parts

Integration by Partial Fractions

Integrating Functions Using Long Division

For more information on each of these, see the corresponding articles.

Let's do a few examples using the Power Rule for Integration.

First, we can use the Power Rule to evaluate integrals with added terms and radicals.

Evaluate the integral

\[\int \sqrt{x} + \frac{17}{\sqrt{\pi}}x^{16} \; dx .\]

**Solution:**

The first thing to notice is that you have two terms in our integral added to each other. By the sum rule for integration, you can write

\[\int \sqrt{x} + \frac{17}{\sqrt{\pi}}x^{16} \; dx = \int \sqrt{x} \; dx + \int \frac{17}{\sqrt{\pi}}x^{16} \; dx.\]

The second term can be integrated by directly applying the power rule with \(a = \frac{17}{\sqrt{\pi}}\) and \(n = 16\). To integrate the first term, use the identity

\[\sqrt{x} = x^{1/2}.\]

Using this identity and applying the power rule, you get that

\[\begin{align}\int \sqrt{x} + \frac{17}{\sqrt{\pi}}x^{16} \; dx &= \int \sqrt{x} \; dx + \int \frac{17}{\sqrt{\pi}}x^{16} \; dx\\&= \int x^{1/2} \; dx + \int \frac{17}{\sqrt{\pi}}x^{16} \; dx\\&= \frac{1}{3/2}x^{3/2} + \frac{17/\sqrt{\pi}}{17}x^{17}\\&= \frac{2}{3}x^{3/2} + \frac{1}{\sqrt{\pi}}x^{17}.\end{align}\]

Let's do another example, this time using negative exponents.

Evaluate

\[\int \frac{1}{x^3} - 4x^{3/2} \; dx.\]

**Solution:**

Since the integrand is the difference of two terms, you can split up the integral using the difference rule for integrals:

\[\int \frac{1}{x^3} - 4x^{3/2} \; dx = \int \frac{1}{x^3} \; dx - \int 4x^{3/2} \; dx\]

To integrate the first term, use the identity

\[\frac{1}{x^3} = x^{-3}.\]

This gives you:

\[\begin{align}\int \frac{1}{x^3} - 4x^{3/2} \; dx &= \int \frac{1}{x^3} \; dx - \int 4x^{3/2} \; dx\\&= \int x^{-3} dx - \int 4x^{3/2} \; dx\\&= \frac{1}{-2}x^{-2} - \frac{4}{5/2} x^{5/2}\\&= -\frac{1}{2}x^{-2} - \frac{8}{5} x^{5/2}.\end{align}\]

Finding new techniques of integration is almost a cottage industry among mathematicians. This is because techniques of integration can be interesting, unexpected, and just flat-out *fun* to mess with. It is beyond the scope of this article to detail every technique of integration known, but we can look at a few examples. In particular, we will look at integrals of inverse functions, techniques for trigonometric integrals, and Feynman's technique of integration.

There is a nice result that gives the integral of any inverse function. (For a refresher, see the article Inverse Functions.)

Given a function \(f(x)\) that has inverse \(f^{-1}(x)\) and antiderivative,

\[\int f^{-1}(y) dy = yf^{-1}(y) - F(f^{-1}(y)) + C.\]

Note that \(f(x)\) that has inverse \(f^{-1}(x)\) and antiderivative \(F(x)\) means that \(\frac{d}{dx}F(x) = f(x)\) .

Here, \(y\) is used instead of \(x\) to emphasize that you are working with inverse functions. This equation can be directly verified by differentiating, using the identity

\[\frac{d}{dy}f^{-1}(y) = \frac{1}{f'(f^{-1}(y))}. \]

With the identity, you get

\[\begin{align}\frac{d}{dy}\left(yf^{-1}(y) - F(f^{-1}(y)) + C\right) &= \frac{d}{dy}\left(yf^{-1}(y)\right) - \frac{d}{dy}\left(F(f^{-1}(y)) + C\right)\\&= f^{-1}(y) + y\left(\frac{1}{f'(f^{-1}(y))}\right) - F'(f^{-1}(y))\frac{1}{f'(f^{-1}(y))}\\&= f^{-1}(y) + y\left(\frac{1}{f'(f^{-1}(y))}\right) - f(f^{-1}(y))\frac{1}{f'(f^{-1}(y))}\\&= f^{-1}(y) + y\left(\frac{1}{f'(f^{-1}(y))}\right) - y\frac{1}{f'(f^{-1}(y))}\\&= f^{-1}(y),\end{align}\]

which is exactly what you wanted to see.

There is also a version of this theorem for definite integrals.

Given a function f with inverse \(f^{-1}\),

\[\int_{f(a)}^{f(b)} f^{-1}(y) dy + \int_a^b f(x) \; dx = bf(b)-af(a).\]

There is a nice visual proof of this fact using properties of inverse functions. First, for simplicity's sake, set \(a = f(a) = 0\) and \(b > 0, \, f(b) > 0\). You can draw the integral \(\int_0^f(b) f(x) \; dx\) like this, where the shaded area is the value of the integral:

You might remember that reflecting the graph of \(f\) about the line \(y=x\) gives the graph of \(f^{-1}\):

You need to find the sum of the integral of the function and its inverse. This is where you can use a nice trick: treat \( f^{-1} \) as a function of \(y\), not as a function of \(x\). Algebraically, this would mean replacing every occurrence of \(x\) with \(y\) and every \(y\) with an \(x\) in the equation for \(f^{-1}\). This is just a change of variables; it doesn't change the integral you are working with at all. Here's the kicker: replacing \(y\) with \(x\) and \(x\) with \(y\) is geometrically equivalent to flipping the graph of \(f^{-1}\) about the line \(y=x\), as you can see in the picture below.

So, as the picture above shows, the sum of our integrals is the same as the area of the rectangle, which is \(bf(b)\).

Now, if you let \(a\) and \(c\) be non-zero, all you are doing geometrically is cutting out a rectangle with area \(af(a)\) from the area you are looking for as you can see in the graph below.

Thus, in general,

\[\int_{f(a)}^{f(b)} f^{-1}(y)dy + \int_a^b f(x) \; dx = bf(b)-af(a).\]

Let's do an example of finding the antiderivative of an inverse function.

Evaluate

\[\int \cos^{-1}(y)dy\]

where \(\cos(x)\)) is considered as a function on the interval \([0,\pi].\)

**Solution:**

The function \(\cos^{-1}(y)\) is the inverse of the function \(\cos(x)\), so you can use the formula

\[\int f^{-1}(y) dy = yf^{-1}(y) - F(f^{-1}(y)) + C.\]

First, note that

\[F(x) = \int \cos(x) \; dx = -\sin(x) + C.\]

So, plugging the functions into the equation above, you get that

\[\int cos^{-1}(y) \; dx = y\cos^{-1}(y) - \sin(\cos^{-1}(y)) + C.\]

You can simplify this expression somewhat by using properties of trigonometric functions. Let \(\theta = \cos^{-1}(x)\). Then

\[\cos(\theta) = x = \frac{x}{1}.\]

As you recall, the cosine of an angle theta can be interpreted in terms of right triangles as the ratio of the adjacent side of the triangle to its hypotenuse.

Using this same triangle, you get that

\[\sin(\cos^{-1}(y)) = \sin(\theta) = \sqrt{1-y^2}.\]

Thus, the expression simplifies to

\[\int cos^{-1}(y) \; dx = y\cos^{-1}(y) - \sqrt{1-y^2} + C.\]

This 'triangle trick' is handy for many integrals. See the article Trigonometric Substitution for more examples.

Let's do an example of finding the definite integral of an inverse function.

Evaluate

\[\int_1^e \ln(y) dy = \int_{e^0}^{e^1} \ln(y) dy.\]

**Solution:**

First, note that \(\ln(y)\) is the inverse of \(e^x\). The integral of \(\ln(y)\) is not necessarily obvious, but we know how to integrate \(e^x\). So, this is a great situation to use properties of inverse functions. Plugging this information into the equation for the definite integral of an inverse function, you get that

\[\begin{align} \int_{e^0}^{e^1} \ln(y) dy + \int_{0}^1 e^x \; dx &= 1(e^1) - 0(e^0) \\ &= e. \end{align}\]

Next, you can integrate to find

\[\begin{align}\int_0^1 e^x \; dx &= e^x \bigg|_{x=0}^{x=1}\\&= e^1 - e^0\\&= e-1.\end{align}\]

Finally, you can solve for

\[\int_{e^0}^{e^1} \ln(y) \; dy. \]

Substituting in what you know,

\[ \int_{e^0}^{e^1} \ln(y) \; dy + e-1 = e, \]

which means that\[ \int_{e^0}^{e^1} \ln(y) \; dy = 1. \]

Trigonometric functions turn up in many integrals and can be quite useful, even in unexpected places. For details on trig substitution or how to integrate trig functions in general, see the articles Trigonometric Substitution, Trigonometric Integrals, and Integrals Resulting in Inverse Trigonometric Functions. Here, you can take a look at Weierstrass Substitution, an interesting technique used to evaluate rational functions of sine and cosine. This technique relies on \(u\) substitution, so it may be helpful to read the article Integration by Substitution before reading this section.

Weierstrass substitution is an elegant method of solving integrals that are rational functions of sine and cosine. The Weierstrass substitution is the substitution \(u = \tan(x/2)\). Using double angle identities, this substitution gives us the formulas:

\[\begin{align} \sin(x) &= \frac{2u}{1+u^2}, \\ \cos(x)&=\frac{1-u^2}{1+u^2}, \\ dx &= \frac{2}{1+u^2} \; du. \end{align}\]

This technique is particularly useful when sine or cosine functions are in the denominator of an integral.

Use Weierstrass substitution to find

\[\int \frac{dx}{\cos(x)}.\]

**Solution:**

First, make the substitution

\[ \begin{align} u &= \tan\left(\frac{x}{2}\right),\\ du &= \frac{1+u^2}{2}dx. \end{align}\]

You can use the equations

\[\sin(x) = \frac{2u}{1+u^2}\]

and

\[dx = \frac{2}{1+u^2} \; du\]

to get that

\[\begin{align}\int \frac{1}{\sin(x)}dx &= \int \frac{1+u^2}{2u}\left(\frac{2}{1+u^2}\right) \; du\\&= \int \frac{1}{u} \; du\\&= \ln|u| + C\\&= \ln\left|\tan\left(\frac{x}{2}\right)\right| + C.\end{align}\]

Feynman's integration technique is an interesting integration technique that is sometimes also referred to as 'differentiating under the integral sign'. Feynman's technique allows you to use differentiation on complicated integrals to obtain an expression that is (hopefully!) easier to integrate.

Feynman's technique is difficult to express succinctly because of how much it varies between different integrals. However, the following steps give at least an outline of what we mean by 'Feynman's technique' for evaluating

\[\int_a^b f(x) dx.\]

Define a function \[I(t) = \int_a^b f(x, t)\] by adding a term \(t\) to the integral. You want to make sure that \(I(0) = 0\) and \[I(c) = \int_a^b f(x) dx\] for some \(c\). The best \(f(x, t)\) to use varies widely depending on which integral you are working with.

For example, if your integral was \[ \int_a^b f(x) dx = \int_a^b x^2 dx,\] you might define \[I(t) = \int_a^b x^t dx= \int_a^b f(x, t) dx.\]

Find \(I'(t)\) by differentiating

*with respect to \(t\)*under the integral sign.In symbols, \[ I'(t) = \frac{d}{dt}\int_a^b f(x, t) dx = \int_a^b \frac{\partial}{\partial t} f(x, t) dx.\]

Using the Fundamental Theorem of Calculus, integrate \(I'(t)\) to find \(I(c)\).

In symbols, \[I(c) = \int_0^c I'(t) dt = \int_a^b f(x) dx\]

Here, the expression \(\frac{\partial}{\partial t}\) just means 'differentiate the expression with respect to \(t\), not with respect to \(x\).' This is called a **partial derivative**; you will see more of these if you take multivariable calculus.

Let's do a couple of examples to illustrate this technique.

Evaluate the integral

\[\int_0^1 \frac{x^3 - 1}{\ln(x)} dx.\]

**Solution:**

To evaluate this integral, you first make the perhaps counterintuitive step of introducing a function of a variable \(t\):

\[I(t) = \int_0^1 \frac{x^t - 1}{\ln(x)}dx.\]

By definition,

\[I(3) = \int_0^1 \frac{x^3 - 1}{\ln(x)},\]

which is the original integral. So, your problem now becomes finding \(I(3)\). To achieve this, start by finding the derivative of our function \(I\):

\[\begin{align}I'(t) &= \frac{d}{dt}\int_0^1 \frac{x^t - 1}{\ln(x)} dx\\&= \int_0^1 \frac{\partial}{\partial t}\frac{x^t - 1}{\ln(x)} dx\\&= \int_0^1 \frac{\partial}{\partial t}\left(\frac{x^t}{\ln(x)} - \frac{1}{\ln(x)}\right) dx\\&= \int_0^1 \frac{\ln(x)x^t}{\ln(x)} dx\\&= \int_0^1 x^t dx\\&= \frac{1}{t+1}x^t \bigg|_{x=0}^{x=1}\\&= \frac{1}{t+1}.\end{align}\]

Here, since you are differentiating with respect to \(t\), you can treat \(x\) as a constant.

Next, since

\[\begin{align} I(0) = \int_0^1 \frac{x^0 - 1}{\ln(x)}dx \\ &= \int_0^1 \frac{1-1}{\ln(x)}dx \\ &= \int_0^1 0 dx \\ &= 0,\end{align}\]

you can use The Fundamental Theorem of Calculus to write:

\[\begin{align}I(3) &= I(3) - I(0)\\&= \int_0^3 I'(t) dt\\&= \int_{0}^3 \frac{1}{t+1} dt\\&= \ln(t+1)\bigg|_{t=0}^{t=3}\\&= \ln(4) - \ln(1)\\&= \ln(4).\end{align}\]

Thus,

\[\int_0^1 \frac{x^3 - 1}{\ln(x)} dx = \ln(4).\]

Let's do one more example.

Evaluate

\[\int_0^{\infty} \frac{\tan^{-1}(x)}{x(1+x^2)} \; dx. \]

See the article Improper Integrals for information on how to solve integrals of this form.

**Solution:**

First, introduce a new parameter \(t\) into the equation. As it turns out, the choice

\[I(t) = \int_0^{\infty} \frac{\tan^{-1}(tx)}{x(1+x^2)} \; dx\]

works well. When \(t = 1\),

\[I(1) = \int_0^{\infty} \frac{\tan^{-1}(x)}{x(1+x^2)} \; dx.\]

Also,

\[\begin{align} I(0) &= \int_0^{\infty} \frac{\tan^{-1}(0)}{x(1+x^2)} \; dx \\ &= \int_0^\infty 0 \; dx \\ &= 0. \end{align}\]

Your next step is to find \(I'(t)\):

\[\begin{align}I'(t) &= \frac{d}{dt}\int_0^{\infty} \frac{\tan^{-1}(tx)}{x(1+x^2)} \; dx\\&= \int_0^{\infty} \frac{\partial}{\partial t}\frac{\tan^{-1}(tx)}{x(1+x^2)} \; dx\\&= \int_0^{\infty} \frac{x}{(1+t^2x^2)x(1+x^2)} \; dx\\&= \int_0^{\infty} \frac{1}{(1+t^2x^2)(1+x^2)} \; dx. \end{align}\]

To continue the integration, use Integration by Partial Fractions:

\[ \begin{align} \int_0^{\infty} \frac{1}{(1+t^2x^2)(1+x^2)} \; dx &= \int_0^{\infty} \frac{-t^2/(1-t^2)}{1 + t^2x^2} \; dx + \int_0^{\infty} \frac{1/(1-t^2)}{1+x^2} \; dx \\&= -\frac{t}{1-t^2}\int_0^{\infty} \frac{t}{1 + t^2x^2} \; dx + \frac{1}{1-t^2}\int_0^{\infty} \frac{1}{1+x^2}\\&= -\frac{t}{1-t^2}\tan^{-1}(tx) + \frac{1}{1-t^2}\tan^{-1}(x)\bigg|_{x=0}^{x=\infty}. \end{align} \]

Now to evaluate the last term, use the fact that the arctangent function approaches \(\frac{\pi}{2}\) as \(x\) approaches infinity, so:

\[ \begin{align} I'(t) &= -\frac{t\pi}{2(1-t^2)} + \frac{\pi}{2(1-t^2)} + \frac{t}{1-t^2}(0) - \frac{1}{1-t^2}(0)\\&= \frac{\pi (t-1)}{2(t^2 - 1)}\\&= \frac{\pi}{2(t+1)}.\end{align}\]

Finally, use the Fundamental Theorem of Calculus to find \(I(1)\):

\[\begin{align}I(1) &= I(1) - I(0)\\&= \int_0^1 I'(t) dt\\&= \int_0^1 \frac{\pi}{2(t+1)} dt\\&= \frac{\pi}{2}\int_0^1\frac{1}{t+1} dt\\&= \frac{\pi}{2}\ln|t+1|\bigg|_{t=0}^{t=1}\\&= \frac{\pi}{2}\left[\ln|2| - \ln|1|\right]\\&= \frac{\pi}{2}\ln(2).\end{align}\]

As in this example, you often need to use Feynman integration with other techniques of integration.

Richard Feynman (1918-1988) was an American theoretical physicist who did significant work in particle physics and quantum mechanics. He was a brilliant physicist with a gift for explaining difficult concepts clearly, elegantly, and concretely. While he did not originate the integration technique that bears his name, he did play a role in popularizing it. Here is what he had to say on the technique:

“I had learned to do integrals by various methods shown in a book that my high school physics teacher Mr. Bader had given me. One day he told me to stay after class. "Feynman," he said, "you talk too much and you make too much noise. I know why. You're bored. So I'm going to give you a book. You go up there in the back, in the corner, and study this book, and when you know everything that's in this book, you can talk again." ... [That book] showed how to differentiate parameters under the integral sign — it’s a certain operation. It turns out that’s not taught very much in the universities; they don’t emphasize it. But I caught on how to use that method, and I used that one damn tool again and again. So because I was self-taught using that book, I had peculiar methods of doing integrals. ... So I got a great reputation for doing integrals, only because my box of tools was different from everybody else’s, and they had tried all their tools on it before giving the problem to me.”^{1}

- Common techniques of integration include the Power Rule for Integrals, Integration by Substitution, Trigonometric Substitution, Integration by Parts, Integration by Partial Fractions, and Integrating Functions Using Long Division.
- The Power Rule for Integration is a rule that 'undoes' the power rule for differentiation.
- Weierstrass substitution is a useful substitution for rational expressions of trigonometric functions.
- The integral of a function can be expressed in terms of its inverse.
- Feynman's technique of integration is a useful technique for complicated integrals that involves differentiating under the integral sign.

- Richard Feynman, Surely You're Joking, Mr. Feynman!, 1985.

More about Techniques of Integration

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