Integrating standard functions

In a nutshell

It is possible to integrate functions other than those of the form $ax^n$ . Knowing some standard integrals will help solve problems quicker.

Standard integrals

Here is a list of integrals you need to know. In it, $n$ and $C$ are constants.

function	integral
$x^n$	$\int x^n \, dx = \dfrac{1}{n+1}x^{n+1}+C, n\ne 1$
$\dfrac1x$	$\int \dfrac1x \, dx = \ln \vert x\vert +C$
$e^x$	$\int e^x \, dx = e^x+C$
$\sin(x)$	$\int \sin(x) \, dx = -\cos(x)+C$
$\cos(x)$	$\int \cos(x) \, dx = \sin(x)+C$
$\sec^2 (x)$	$\int \sec^2(x) \, dx = \tan(x)+C$
$\cosec^2(x)$	$\int \cosec^2 (x) \, dx = -\cot(x)+C$
$\sec(x) \tan(x)$	$\int \sec(x) \tan(x) \, dx = \sec(x)+C$
$\cosec(x)\cot(x)$	$\int \cosec(x)\cot(x) \, dx = -\cosec(x)+C$

Note: The trigonometric integrals assume that $x$ is in radians, not degrees.

Example 1

Compute the following integrals:

i) $\int \dfrac{1}{4x} \, dx$ 

ii) $\int_\frac{\pi}{2}^\frac{3\pi}{2} \left(\dfrac{-4\cot(x)}{\sin(x)}-\cos(x) \right) \, dx$ 

Part i):

Use the fact that $\int kf(x)\, dx = k\int f(x)\, dx$ to turn the integral into one of the standard integrals.

$\begin{aligned} \int\dfrac{1}{4x} \, dx &= \int \dfrac{1}{4} \times \dfrac1x \, dx\\&=\dfrac14 \int \dfrac1x \, dx\\&=\dfrac14 \ln \vert x\vert +C \end{aligned}$

$\int \dfrac{1}{4x}\, dx= \underline{\dfrac14 \ln \vert x\vert +C}$

Part ii):

Use the fact that $\int (f(x)+g(x))\, dx = \int f(x)\, dx + \int g(x)\, dx$ together with the procedure for definite integration.

$\begin{aligned} \int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} \left(\dfrac{-4\cot(x)}{\sin(x)}-\cos(x) \right) \, dx &= \int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} \left(-4\cot(x) \times\dfrac{1}{\sin(x)} - \cos(x)\right)\, dx\\&=\int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} \left(-4\cot(x)\cosec(x) -\cos(x)\right) \, dx\\&=-4\int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} \cot(x) \cosec(x) \, dx - \int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} \cos(x)\, dx\\&=\left[-4(-\cosec(x)) - (\sin(x))\right]_{\frac{\pi}{2}}^{\frac{3\pi}{2}}\\&=\left[4\cosec(x) - \sin(x)\right]_{\frac{\pi}{2}}^{\frac{3\pi}{2}}\\&=\left(\dfrac{4}{\sin \left(\frac{3\pi}{2} \right)} - \sin\left(\frac{3 \pi}{2}\right)\right) - \left(\dfrac{4}{\sin \left(\frac{\pi}{2}\right)} - \sin\left(\frac{ \pi}{2}\right)\right)\\&=\left(\dfrac{4}{-1} - (-1)\right) - \left(\dfrac{4}{1} - (1)\right)\\&=(-3)-(3)\\&=-6 \end{aligned}$

$\int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} \left(\dfrac{-4\cot(x)}{\sin(x)}-\cos(x) \right) \, dx= \underline{-6}$