2 A table of integrals

We could use a table of derivatives to find integrals, but the more common ones are usually found in a ‘Table of Integrals’ such as that shown below. You could check the entries in this table using your knowledge of differentiation. Try this for yourself.

When dealing with the trigonometric functions the variable $x$ must always be measured in radians and not degrees. Note that the fourth entry in the Table, for $x^n$ , is valid for any value of $n$ , positive or negative, whole number or fractional, except $n=-1$ . When $n=-1$ use the fifth entry in the Table.

Example 1

Use Table 1 to find the indefinite integral of $x^7$ : that is, find $\int \; <!--nolimits\; -->x^{7}dx$

Solution

From Table 1 note that $\int \; <!--nolimits\; -->x^{n}dx=\frac{x^{n+1}}{n+1}+c$ . In words, this states that to integrate a power of $x$ , increase the power by 1, and then divide the result by the new power. With $n=7$ we find

$\int \; <!--nolimits\; -->x^{7}dx=\frac{1}{8}{x}^8+c$

Example 2

Find the indefinite integral of $cos5x$ : that is, find $\int \; <!--nolimits\; -->cos5xdx$

Solution

From Table 1 note that $\int \; <!--nolimits\; -->coskxdx=\frac{sinkx}{k}+c$

With $k=5$ we find $\int \; <!--nolimits\; -->cos5xdx=\frac{1}{5}sin5x+c$

In Table 1 the independent variable is always given as $x$ . However, with a little imagination you will be able to use it when other independent variables are involved.

Example 3

Find $\int \; <!--nolimits\; -->cos5tdt$

Solution

We integrated $cos5x$ in the previous example. Now the independent variable is $t$ , so simply use Table 1 and replace every $x$ with a $t$ . With $k=5$ we find

$\int \; <!--nolimits\; -->cos5tdt=\frac{1}{5}sin5t+c$

It follows immediately that, for example,

$\int \; <!--nolimits\; -->cos5\omega d\omega =\frac{1}{5}sin5\omega +c,\int \; <!--nolimits\; -->cos5udu=\frac{1}{5}sin5u+c$ and so on.

Example 4

Find the indefinite integral of $\frac{1}{x}$ : that is, find $\int \; <!--nolimits\; -->\frac{1}{x}dx$

Solution

This integral deserves special mention. You may be tempted to try to write the integrand as $x^{-1}$ and use the fourth row of Table 1. However, the formula $\int \; <!--nolimits\; -->x^{n}dx=\frac{x^{n+1}}{n+1}+c$ is not valid when $n=-1$ as Table 1 makes clear. This is because we can never divide by zero. Look to the fifth entry of Table 1 and you will see $\int \; <!--nolimits\; -->x^{-1}dx=ln\left|x\right|+c$ .

Example 5

Find $\int \; <!--nolimits\; -->12dx$ and $\int \; <!--nolimits\; -->12dt$

Solution

In this Example we are integrating a constant, 12. Using Table 1 we find

$\int \; <!--nolimits\; -->12dx=12x+c$ Similarly $\int \; <!--nolimits\; -->12dt=12t+c$ .

Task!

Find $\int \; <!--nolimits\; -->t^{4}dt$

Answer

Find $\int \; <!--nolimits\; -->\frac{1}{x^5}dx$ using the laws of indices to write the integrand as $x^{-5}$ and then use Table 1:

Answer

Task!

Find $\int \; <!--nolimits\; -->{\text{e}}^{-2x}dx$ using the entry in Table 1 for integrating ${\text{e}}^{kx}$ :

Answer

Exercises

1. Integrate each of the following functions with respect to $x$ :
   1. $x^9$ ,
   2. $x^{1∕2}$ ,
   3. $x^{-3}$ ,
   4. $1∕x^4$ ,
   5. 4,
   6. $\sqrt{x}$ ,
   7. ${\text{e}}^{4x}$
2. Find
   1. $\int \; <!--nolimits\; -->t^{2}dt$ ,
   2. $\int \; <!--nolimits\; -->6dt$ ,
   3. $\int \; <!--nolimits\; -->sin3tdt$ ,
   4. $\int \; <!--nolimits\; -->{\text{e}}^{7t}dt$ .

Answer