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Integration Tables

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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 T****ables.**

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.

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?

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.

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!

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 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.

More about Integration Tables

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