Integration Basics

Finding definite and indefinite integrals of $x^n$ for $n$ rational, $n \neq 1$, and related sums and differences, including expressions which require simplification prior to integrating.

For example: $\int (x + 2)^2 \mathrm{d}x$ and $\int \frac{(3x - 5)^2}{x^2} \mathrm{d}x$

In the TMUA/ESAT we expect you to be able to integrate sums of terms in powers of $x$ using the rule:

$\int k x^n \mathrm{d}x = \frac{k x^{n+1}}{n + 1} + c \quad n \neq -1$

Where $k$ and $c$ are a real constant and $n$ is any real number except $-1$. When $n = -1$ the answer falls outside the scope of TMUA/ESAT. (You might have seen: $\int \frac{1}{x} \mathrm{d}x \ln|x| + c$. Note that $\ln|x|$ is the natural logarithm of $x$ and this topic is NOT on the TMUA/ESAT.)

Any integration that you are required to do in the TMUA/ESAT, if it does not require symmetry or other arguments, will be an integration of sums of powers of $x$. We will not ask question that require more sophisticated methods such as substitution or integration by parts etc., and we are careful to make sure every question we might ask involving integration can be answered equally efficiently using the basic methods outlined here, even if it turns out more advanced methods could also be applied. (We are careful in the TMUA/ESAT to ensure that knowing more advanced techniques does not give anyone undue advantage.)

In the TMUA/ESAT the expression you are given might not look like powers of $x$ but it will be possible to simplify it into such a sum. Here is an example:

$\begin{aligned} \int (x - 2)^2 \mathrm{d}x &= \int x^2 - 4x + 4 \mathrm{d}x = \int x^2 \mathrm{d}x - \int 4x \mathrm{d}x + \int 4 \mathrm{d}x \\ &= \frac{x^3}{3} - \frac{4x^2}{2} + 4x + c \end{aligned}$

Here it is useful to notice something about integration that is often taken for granted without much thought: when you integrate a simple sum, you can integrate term by term and add [or subtract] the individual answers. It is always worth thinking very carefully about the properties of the mathematics you meet to ensure you do not inadvertently perform mathematical moves that might seem right but which are, in fact, invalid. (For instance, it might be tempting to write $f(x + 3) = f(x) + f(3)$ or $f(x^2) = [f(x)]^2$ but these are not generally true. Or you might be tempted to do this [which is very wrong, so don't do it]: $\int \frac{f(x)}{g(x)} \mathrm{d}x = \frac{\int f(x) \mathrm{d}x}{\int g(x) \mathrm{d}x}$)