Select your language

Suggested languages for you:
Log In Start studying!
StudySmarter - The all-in-one study app.
4.8 • +11k Ratings
More than 3 Million Downloads
Free
|
|

All-in-one learning app

  • Flashcards
  • NotesNotes
  • ExplanationsExplanations
  • Study Planner
  • Textbook solutions
Start studying

Integration Tables

Integration Tables

One of the greatest achievements of humankind is writing. Thanks to it, our knowledge is preserved through generations by the means of books, so we do not need to spend time and energy rediscovering everything!

One of the most important operations in calculus is Integration, which can be time-consuming. Luckily, just like the information humanity has gotten through the ages is contained in books, lots of integrals are stored in Integration Tables.

Method for Using Integration Tables

Integration can be a cumbersome operation. You first need to know what integration method is more suitable for a given integral. After that comes the operation itself. Who knows, maybe you will need to do Integration by Parts several times! This would be time-consuming and complicated.

Rather than going through this trial, it is easier to use an Integration Table.

But how can you use a table to integrate a function? Integration Tables contain summarized formulas for specific integrals. What is important is that you identify the variables and constants that are present in each formula.

Here is a quick example. Consider the integral

\[ \int \sin{3x} \, \mathrm{d}x.\]

To solve this integral you need to do Integration by Substitution by letting

\[u=3x.\]

You also need to write the differential \( \mathrm{d}x \) in terms of \( u,\) which can be done first by differentiating

\[ \frac{\mathrm{d}u}{\mathrm{d}x}=3,\]

multiplying this derivative by \( \mathrm{d}x,\)

\[ \mathrm{d}u=3\,\mathrm{d}x,\]

and isolating \( \mathrm{d}x,\) so

\[\mathrm{d}x=\frac{1}{3}\mathrm{d}u.\]

You can now write the original integral in terms of \(u\) by replacing every instance of \( x \) by its equivalent in \( u,\) and every instance of \( \mathrm{d}x \) by its equivalent in \( \mathrm{d}u,\) that is

\[ \begin{align} \int \sin{3x} \, \mathrm{d}x &= \int (\sin{u})\left(\frac{1}{3}\mathrm{d}u\right) \\ &= \frac{1}{3}\int \sin{u} \, \mathrm{d}u, \end{align}\]

which is an integral that has a common formula which you can check in our Trigonometric Integrals article, that is

\[\int \sin{u} \, \mathrm{d}u = -\cos{u} + C.\]

Knowing this, you can write the integral

\[ \int \sin{3x}\,\mathrm{d}x = \frac{1}{3} \left( -\cos{u} + C \right),\]

and then undo the substitution. Usually, the integration constant is added at the end, so

\[ \begin{align} \int \sin{3x}\,\mathrm{d}x &= \frac{1}{3} \left( -\cos{3x} \right) + C \\ &= -\frac{1}{3}\cos{3x}+C. \end{align}\]

In the above example, by looking at an Integration Table about trigonometric functions you will most likely find a formula like

\[ \int \sin{ax} \, \mathrm{d}x = -\frac{1}{a}\cos{ax}+C.\]

In that case you do not need to do any \(u-\)substitution, but you have to identify that \( a=3.\)

\[ \begin{align} \int \sin{ax}\,\mathrm{d}x &= \frac{1}{a} \left( -\cos{ax} \right) + C \\ \int \sin{3x} \, \mathrm{d}x &= -\frac{1}{3}\cos{3x}+C. \end{align}\]

The main idea of using Integration Tables is to identify which integral from the table has the same form as the one you are trying to solve. The integral given in the table is already solved, so you can use it as a formula.

You can distinguish between variables and constants by looking at the differential of the integral. The variable of integration is the same variable present in the differential, and usually \(x\) or \(u\) are used. The rest of the letters you find are most likely constants, and usually \(a,\) \(b,\) \(k,\) \(n,\) and \(m\) are chosen.

Since there are a lot of different integrals, Integration Tables are usually broken down according to which kind of functions are involved. Here we will take a look at some examples of the most common Integration Tables.

Integration Tables for Exponential Functions

The integration of exponential functions usually need you to integrate by parts a several amount of times. Rather than doing this, you can always look at Integration Tables. These might contain some of the following formulas:

\[ \begin{align}&\int e^{bx}\,\mathrm{d}x = \frac{1}{b}e^{bx}+C\\[0.5em] &\int a^{bx}\,\mathrm{d}x = \frac{1}{b \ln{a}}a^{bx}+C \quad \text{ for }\quad a>0, a\neq 1 \\[0.5em] &\int xe^{bx}\,\mathrm{d}x= \frac{e^{bx}}{b^2}(bx-1)+C\\[0.5em]&\int x^2e^{bx}\,\mathrm{d}x= e^{bx}\left( \frac{x^2}{b}-\frac{2x}{b^2}+\frac{2}{b^3}\right)+C \\[0.5em]&\int xe^{bx}\,\mathrm{d}x = \frac{1}{2b}e^{bx^2}+C \\[0.5em]&\int xe^{-bx}\,\mathrm{d}x=-\frac{1}{2b}e^{-bx^2}+C\end{align}\]

Evaluate the integral

\[ \int x^2 e^{5x}\,\mathrm{d}x.\]

Answer:

You should begin by looking for an integral that looks like the one you are trying to solve. From the integrals given previously you should focus on

\[ \int x^2e^{bx}\,\mathrm{d}x= e^{bx}\left( \frac{x^2}{b}-\frac{2x}{b^2}+\frac{2}{b^3}\right)+C, \]

where you need to identify that \( b=5.\) Knowing this, you can substitute \( b\) in the above formula an obtain

\[ \begin{align} \int x^2e^{5x}\,\mathrm{d}x &= e^{5x}\left( \frac{x^2}{5}-\frac{2x}{5^2}+\frac{2}{5^3}\right)+C \\[0.5em] &= e^{5x}\left( \frac{x^2}{5}-\frac{2x}{25}+\frac{2}{125}\right)+C. \end{align}\]

Pretty straightforward, right?

Integration Tables for Trigonometric Functions

It might be hard to remember all the antiderivatives of the main trigonometric functions, not to mention some special cases when their powers are also involved. Here are some of the most commonly used formulas that you can find in different Integration Tables:

\[ \begin{align}&\int \sin{ax}\,\mathrm{d}x=-\frac{1}{a}\cos{ax}+C \\[0.5em]&\int \sin^2{ax}\,\mathrm{d}x=\frac{x}{2}-\frac{1}{2a}(\sin{ax})(\cos{ax})+C=\frac{x}{2}-\frac{1}{4a}\sin{2ax}+C\\[0.5em]&\int \cos{ax}\,\mathrm{d}x=\frac{1}{a}\sin{ax}+C \\[0.5em]&\int \cos^2{ax}\,\mathrm{d}x=\frac{x}{2}+\frac{1}{2a}(\sin{ax})(\cos{ax})+C=\frac{x}{2}+\frac{1}{4a}\sin{2ax}+C\\[0.5em]&\int \tan{ax}\,\mathrm{d}x=-\frac{1}{a}\ln{\left| \cos{ax} \right|}+C=\frac{1}{a}\ln{\left| \sec{ax} \right|}+C \\[0.5em]&\int \tan^2{ax}\,\mathrm{d}x= \frac{\tan{ax}}{a}-x+C \\[0.5em]&\int (\sin{ax})(\cos{ax})\,\mathrm{d}x=\frac{1}{2a}\sin^2{ax}+C=-\frac{1}{2a}\cos^2{ax}+C\end{align}\]

Please note that in some of the above formulas there are two different ways of writing the integrals. These are related by trigonometric identities, so either one is fine.

Evaluate the integral

\[ \int \cos^2{7x}\,\mathrm{d}x.\]

Answer:

Once again, you should look in a table for an integral that looks like the above. Note that it is the integral of the cosine function squared, so

\[ \int \cos^2{ax}\,\mathrm{d}x=\frac{x}{2}+\frac{1}{2a}(\sin{ax})(\cos{ax})+C \]

should come in handy. In your case \( a=7,\) so

\[ \begin{align} \int \cos^2{7x}\,\mathrm{d}x &= \frac{x}{2}+\frac{1}{2\cdot 7}(\sin{7x})(\cos{7x})+C \\ &= \frac{x}{2}+\frac{1}{14}(\sin{7x})(\cos{7x}) +C. \end{align}\]

Please note that for this integral you could have also used

\[ \int \cos^2{ax}\,\mathrm{d}x= \frac{x}{2}+\frac{1}{4a}\sin{2ax}+C.\]

Both formulas are related by a trigonometric identity and the integration constant.

Integration Tables for Other Formulas

There is for sure a wide variety of integrals. Some might involve Trigonometric Substitution, while others might require Partial Fractions decomposition. Here are more of the most common formulas shown in Integration Tables:

\[\begin{align}&\int \frac{1}{\sqrt{a^2-x^2}}\mathrm{d}x=\arcsin{\frac{x}{a}}+C \\[0.5em]&\int \frac{1}{a^2+x^2}\mathrm{d}x=\frac{1}{a}\arctan{\frac{x}{a}}+C\\[0.5em]&\int \frac{1}{x\sqrt{x^2-a^2}}\mathrm{d}x=\frac{1}{a}\mathrm{arcsec}{\,\frac{|x|}{a}}+C\\[0.5em]&\int \sqrt{x^2 \pm a^2}\,\mathrm{d}x= \frac{1}{2}x\sqrt{x^2\pm a^2}\pm\frac{1}{2}a^2\ln{\left| x+\sqrt{x^2 \pm a^2} \right|}+C\end{align}\]

Evaluate the integral

\[ \int \frac{1}{\sqrt{9-x^2}} \, \mathrm{d}x.\]

Answer:

This time you might struggle to find which formula to use because you might not find a formula that includes

\[\frac{1}{\sqrt{a-x^2}}.\]

There is one that is close enough, that is

\[ \int \frac{1}{\sqrt{a^2-x^2}}\mathrm{d}x=\arcsin{\frac{x}{a}}+C, \]

you only have to write \( 9 \) as the square of another number, in this case \( 3,\) so

\[ \int \frac{1}{\sqrt{9-x^2}} \mathrm{d}x = \int \frac{1}{\sqrt{3^2-x^2}} \mathrm{d}x.\]

This way you can identify that \( a=3,\) so

\[ \int \frac{1}{\sqrt{9-x^2}}\mathrm{d}x=\arcsin{\frac{x}{3}}+C. \]

Sometimes you will need to play close attention to your integrals to rewrite them like the formulas given in a table.

Evaluate the integral

\[ \int \sqrt{x^2-3}\,\mathrm{d}x.\]

Answer:

In this case, to use the formula

\[ \begin{align} \int \sqrt{x^2 \pm a^2}\,\mathrm{d}x &= \frac{1}{2}x\sqrt{x^2\pm a^2} \\ & \quad \pm\frac{1}{2}a^2\ln{\left| x+\sqrt{x^2 \pm a^2} \right|}+C \end{align} \]

you need to identify \( a, \) and you also need to identify whether to use the plus or the minus sign.

Note that even though \( 3 \) is not a perfect square, it is still \( \sqrt{3}\) squared, so \( a=\sqrt{3}.\) Since your integral uses a minus sign, which is below the plus sign in \( \pm,\) you should use every sign that is below in the formula, so

\[ \begin{align} \int \sqrt{x^2 - a^2}\,\mathrm{d}x &= \frac{1}{2}x\sqrt{x^2 - a^2}\\ & \quad -\frac{1}{2}a^2\ln{\left| x+\sqrt{x^2 - a^2} \right|}+C.\end{align} \]

Finally, substitute \( a \) into the formula, that is

\[ \begin{align} \int \sqrt{x^2-3}\,\mathrm{d}x &= \frac{1}{2}x\sqrt{x^2-\left(\sqrt{3}\right)^2} \\ & \quad -\frac{1}{2}\left( \sqrt{3}\right)^2 \ln{\left| x+\sqrt{x^2-\left( \sqrt{3} \right) ^2} \right|} + C \\[0.5em] &=\frac{1}{2}x\sqrt{x^2-3} - \frac{3}{2}\ln{\left| x+\sqrt{x^2-3} \right|}. \end{align} \]

Imagine having to do the above integral without a table!

Integration Tables for the Gaussian Function

Not all functions have antiderivatives, that is, you will not always be able to find a "nice" function to solve an integral. Such is the case of the Gaussian Function

\[ f(x)= e^{-x^2}.\]

No matter what integration method you try to use, you just will not be able to find its antiderivative!

The above function is very important in Statistics, and evaluating its definite integral

\[ \int_0^b e^{-x^2}\,\mathrm{d}x\]

becomes extremely relevant. Since you cannot use the Fundamental Theorem of Calculus to evaluate the above integral, it is evaluated numerically instead, and its values are organized in tables. For more information about this check out our article about the Normal Distribution!

Integration Tables - Key takeaways

  • Integration Tables contain summarized formulas for specific integrals.
  • The main idea of using Integration Tables is to identify an integral that has the same form as one from the table.
  • There are Integration Tables for exponential functions, trigonometric functions, and even more! You should look for a table that best suits the integral you need to solve.

Frequently Asked Questions about Integration Tables

To use an integration table you need to identify which integral from the table has the same form as the one you are trying to solve. The integral given in the table is already solved, so you can use it as a formula.

An integration table is a list containing common integrals and their solution. These are usually organized according to which types of functions are involved, like exponential integration tables or trigonometric integration tables.

To solve an integral using a table you need to identify which integral from the table has the same form as the one you are trying to solve. The integral given in the table is already solved, so you can use it as a formula.

Once you have solved one type of integral you can write down the answer and keep it. By solving and organizing more integrals you can build an integration table according to your needs.

Final Integration Tables Quiz

Question

Consider the following integral
\[ \int \sin^2{5x} \, \mathrm{d}x.\]
Which kind of integration table should you use to solve it?

Show answer

Answer

An integration table with trigonometric functions.

Show question

Question

Suppose you want to use a formula for

\[ \int a^{bx}\,\mathrm{d}x\] 
to solve 
\[ \int 5^{3x}\,\mathrm{d}x.\]
What are the values of \( a \) and \( b \)?

Show answer

Answer

\( a=5 \) and \( b=3.\)

Show question

Question

Suppose you want to use a formula for

\[ \int \sin^2{ax}\,\mathrm{d}x\]
to solve
\[ \int \sin^2{3x}\,\mathrm{d}x.\]
What is the value of \( a\) ?

Show answer

Answer

\( a=3.\)

Show question

Question

Suppose you want to use a formula for
\[ \int \sqrt{a^2-x^2}\,\mathrm{d}x\]
to solve
\[ \int \sqrt{4-x^2}\,\mathrm{d}x.\]
What is the value of \( a\) ?

Show answer

Answer

\( 2.\)

Show question

Question

Suppose you want to use a formula for

\[ \int \sqrt{a^2-x^2}\,\mathrm{d}x\]
to solve
\[ \int \sqrt{7-x^2}\,\mathrm{d}x.\]
What is the value of \( a \)?

Show answer

Answer

\( a=\sqrt{7}.\)

Show question

Question

Suppose you want to use a formula for
\[ \int \sin{ax}\,\mathrm{d}x\]
to solve
\[ \int \sin{\left( \frac{x}{5} \right)}\,\mathrm{d}x.\]
What is the value of \( a\) ?

Show answer

Answer

\( \frac{1}{5}.\)

Show question

Question

Consider the formula

\[ \int \frac{x}{\sqrt{x^2\pm a^2}} \mathrm{d}x = \sqrt{x^2 \pm a^2}+C.\]
Use the above formula to evaluate
\[ \int \frac{x}{\sqrt{x^2-9}} \mathrm{d}x.\]

Show answer

Answer

\[ \sqrt{x^2-9} + C. \]

Show question

Question

Consider the formula
\[ \int \frac{x}{\sqrt{x^2 \pm a^2}} \,\mathrm{d}x = \sqrt{x^2 \pm a^2}+C.\]
Use the above formula to evaluate
\[ \int \frac{x}{\sqrt{x^2+4}}\,\mathrm{d}x.\]

Show answer

Answer

\[ \sqrt{x^2+4} + C.\]

Show question

Question

Consider the following integral
\[ \int 3^{2x} \, \mathrm{d}x.\]

Which kind of integration table should you use to solve it?

Show answer

Answer

An integration table with exponential functions.

Show question

Question

Is it possible to have more than one formula for the same integral?

Show answer

Answer

Yes. This usually happens with trigonometric integrals.

Show question

More about Integration Tables
60%

of the users don't pass the Integration Tables quiz! Will you pass the quiz?

Start Quiz

Discover the right content for your subjects

No need to cheat if you have everything you need to succeed! Packed into one app!

Study Plan

Be perfectly prepared on time with an individual plan.

Quizzes

Test your knowledge with gamified quizzes.

Flashcards

Create and find flashcards in record time.

Notes

Create beautiful notes faster than ever before.

Study Sets

Have all your study materials in one place.

Documents

Upload unlimited documents and save them online.

Study Analytics

Identify your study strength and weaknesses.

Weekly Goals

Set individual study goals and earn points reaching them.

Smart Reminders

Stop procrastinating with our study reminders.

Rewards

Earn points, unlock badges and level up while studying.

Magic Marker

Create flashcards in notes completely automatically.

Smart Formatting

Create the most beautiful study materials using our templates.

Sign up to highlight and take notes. It’s 100% free.