Basics
This section introduces the concept of antidifferentiation, linearity, power rule and trigonometric identities.
Definition
Antiderivatives are the inverse of the differentiation. Given some function f(x) where
f^\prime(x) = g(x)
the antiderivative of g(x) is f(x). In general, the antiderivative of a function g(x) is a separate function f(x) whose we must differentiate to obtain g(x). This yields our definition.
These antiderivatives are also called indefinite integrals, or just integrals for short. For any function g(x), there exists an infinite set of antiderivatives f(x) which only vary by some constant. Thus we write
\int g(x) \, dx = f(x) + C
where f(x) is some antiderivative of g(x) and C is any real number. Antiderivatives do not always exists for any arbitrary function g(x), and when they do exist, they may not be possible to express in elementary terms.
From this point on, we will refer almost exclusively to antiderivatives as integrals. We will call functions integrable if their antiderivative exists.
Linearity
Integrals have a property called linearity. This means they are closed under linear transformations of functions like addition and scaler multiplication. For example
\int f(x) + g(x) \,dx = \int f(x) \,dx + \int g(x) \,dx.
Or, given some constant C
\int C f(x) \,dx = C \int f(x) \,dx.
This is presented in the following theorem without proof.
Inverse power rule
One of the basic rules of differentiation is that of power rule.
\frac{d}{dx} x^m = mx^{m-1}
Suspecting this will be a useful property, we can derive an expression for the inverse power rule from the definition of an integral.
\int mx^{m-1} dx = x^m
Apply linearity and the substituion n = m-1
\int x^{n} dx = \frac{x^{n+1}}{n+1}
Edge case
One may notice that when n=-1, inverse power rules gives us an undefined result
\int x^{-1}\,dx = \frac{1}{0} + C.
For this particular value of n, we get the expression
\int x^{-1}\,dx = \ln|x| + C.
Note the absolute value bars. This is because the function \ln is not defined for negative values of x, where our integrand is.
Common identities
This section will cover many commonly used integration identities without much explanation.
Trig identities
Trigonometry is a huge part of the integration bee, and will repeatedly appear in almost every section of this guide. As such it is crucial to have a complete understanding of these functions and how they interact.
The integrals below are the direct inverse of differentiating the six primary trig functions.
\begin{array}{ll} \displaystyle\int{{\cos x\,dx}} = \sin x + C \hspace{0.5in} & \displaystyle\int{{\sin x\,dx}} = - \cos x + C\\ \displaystyle\int{{{{\sec }^2}x\,dx}} = \tan x + C \hspace{0.5in} & \displaystyle\int{{{{\csc }^2}x\,dx}} = - \cot x + C\\ \displaystyle\int{{\sec x\tan x\,dx}} = \sec x + C \hspace{0.5in} & \displaystyle\int{{\csc x\cot x\,dx}} = - \csc x + C \end{array}It is helpful (for me at least) to memorize the left column of the table, and then derive the right side by replacing each function with its ‘compliment’, and then negating the result.
Inverse Trig identities
We can use special formulas to differentiate inverse functions. By differentiating \arcsin(x) and \arctan(x), we get
\int \frac{1}{x^2 + 1} \, dx = \arctan(x) + C \hspace{0.5in} \int \frac{1}{\sqrt{1-x^2}}\,dx = \arcsin(x) + C
It is worth noting that
\frac{d}{dx}\arccos(x) = -\frac{1}{1-x^2}.
So the second integral can also be solved
\int \frac{1}{\sqrt{1-x^2}}\,dx = -\arccos(x) + C.
This is because \arcsin and -\arccos differ by a constant (\frac{\pi}{2}).
Exponential and logarithmic functions
By once again inverting common derivative rules, we get the following identities.
\int e^x \, dx = e^x + C \hspace{0.5in} \int a^x \,dx = \frac{a^x}{\ln{x}} + C
Examples
Some example problems from various websites