3 Some rules of integration

To enable us to find integrals of a wider range of functions than those normally given in a table of integrals we can make use of the following rules.

3.1 The integral of $kf$ ( $x$ ) where $k$ is a constant

A constant factor in an integral can be moved outside the integral sign as follows:

Example 6

Find the indefinite integral of $11x^2$ : that is, find $\int \; <!--nolimits\; -->11x^{2}dx$

Solution

$\int \; <!--nolimits\; -->11x^{2}dx=11\int \; <!--nolimits\; -->x^{2}dx=11\left(\frac{x^3}{3}+c\right)=\frac{11x^3}{3}+K$ where $K$ is a constant.

Example 7

Find the indefinite integral of $-5cosx$ ; that is, find $\int \; <!--nolimits\; -->-5cosxdx$

Solution

$\int \; <!--nolimits\; -->-5cosxdx=-5\int \; <!--nolimits\; -->cosxdx=-5\left(sinx+c\right)=-5sinx+K$ where $K$ is a constant.

3.2 The integral of $f\left(x\right)+g\left(x\right)$ and of $f\left(x\right)-g\left(x\right)$

When we wish to integrate the sum or difference of two functions, we integrate each term separately as follows:

Example 8

Find $\int \; <!--nolimits\; -->\left(x^3+sinx\right)dx$

Solution

$\int \; <!--nolimits\; -->\left(x^3+sinx\right)dx=\int \; <!--nolimits\; -->x^{3}dx+\int \; <!--nolimits\; -->sinxdx=\frac{1}{4}{x}^4-cosx+c$

Note that only a single constant of integration is needed.

Task!

Find $\int \; <!--nolimits\; -->\left(3t^4+\sqrt{t}\right)dt$

Use Key Points 1 and 2:

Answer

Task!

The hyperbolic sine and cosine functions, $sinhx$ and $coshx$ , are defined as follows:

$sinhx=\frac{{\text{e}}^x-{\text{e}}^{-x}}{2}coshx=\frac{{\text{e}}^x+{\text{e}}^{-x}}{2}$

Note that they are combinations of the exponential functions ${\text{e}}^x$ and ${\text{e}}^{-x}$ .

Find the indefinite integrals of $sinhx$ and $coshx$ .

Answer

Further rules for finding more complicated integrals are dealt with in subsequent Sections.