Integration Formula

Integration Formula definition

It might sound absurd at first, that we can have formulas for integrals straight away. But there is a catch, we can only have formulas for relatively simple integrals, not for more composite functions.

Let us start with the very basics, the fundamental formulas of integration:

Basic Integration formulas

The following formulas we are going to talk about, are the very fundamental formulas, upon which other formulas are eventually built.

Hence, it would be a good practice to remember them by heart:

$$ \begin{aligned} &\int x^n \, \mathrm{d}x = \frac{x^{n+1}}{n+1}+c \\ &\int e^x \, \mathrm{d}x=e^x+c \\ &\int a^x \, \mathrm {d}x=\frac{a^x}{\log_e a}+c \\&\int \frac{1}{x} \, \mathrm{d}x=\log_e x +c \end{aligned}$$

Integration formulas of Trigonometric Functions

We just saw integration formulas for functions such as the linear function, quadratic functions, exponential function, logarithmic functions, and also functions like \(\frac{1}{1+x}, \ \frac{1}{1+x^2}\) and so on.

But there is a whole class of important functions that we have not addressed yet, the Trigonometric Functions. There are some basic formulas one needs to remember for trigonometric functions, from which we can solve more complicated trigonometric functions. These Integration formulas are as follows:

$$ \begin{aligned} &\int \cos x \, \mathrm{d}x=\sin x+c \\ &\int \sin x \, \mathrm{d}x=-\cos x+c \\ &\int \tan x \, \mathrm{d}x=-\log_e |\cos x|+c \\ &\int \cot x \, \mathrm{d}x=\log_e |\sin x|+c \\ &\int \sec^2 x \, \mathrm{d}x=\tan x +c \\ &\int \csc x \, \mathrm{d}x=\log_e |\frac{\tan x}{2}|+c \\ &\int \sec x \tan x \, \mathrm{d}x=\sec x+c \\ &\int \csc^2 x \, \mathrm{d}x=-\cot x+c \\ &\int \csc x \cot x \, \mathrm{d}x=-\cot x+c \end{aligned}$$

Definite integral formula

As you may have noticed, the integration we saw are all related to indefinite integrals. But what about definite integrals? The integrals of all the functions remain unchanged, the only thing introduced are the limits of integration.

Below are some formulas, properties essentially, that are crucial while doing definite integration.

$$ \begin{aligned} &\int_a^b f(x) \, \mathrm{d}x=F(b)-F(a) \ ; \text{where} \ \int f(x) \, \mathrm{d}x=F(x)+c \\ &\int_a^b f(x) \, \mathrm{d}x=-\int_b^a f(x) \, \mathrm{d}x \\ &\int_a^b f(x) \, \mathrm{d}x=\int_a^b f(a+b-x) \, \mathrm{d}x \\ &\int_{-a}^a f(x) \, \mathrm{d}x=2\int_0^a f(x)\, \mathrm{d}x\ ; \text{where} \ f(x) \ \text{is an even function} \\ &\int_{-a}^a f(x) \, \mathrm{d}x=0 \ ; \text{where} \ f(x) \ \text{is an odd function} \\ \end{aligned}$$

Integration formula Chart

While solving integrals, we need to know a multitude of different integration formulas, without which, solving integrals will be impossible (unless you cheat using WolframAlpha or something!).

Below is a list of almost all the integration formulas one shall need in order to solve more complicated integrals, which comprise different composite functions. This will include formulas you already saw earlier (trigonometric, fundamental) alongside inverse trigonometric formulas, hyperbolic formulas, and many more. It is a handy chart one should have while doing integrals:

Fundamental Integration Formulas chart

Let us revisit some of the fundamental integration formulas as a chart:

$$ \begin{aligned} &\int x^n\, \mathrm{d} x=\frac{x^{n+1}}{n+1}+c \\ &\int e^x \, \mathrm{d}x=e^x+c \\ &\int a^x \, \mathrm{d}x=\frac{a^x}{\log_e a}+c \\ &\int \frac{1}{x} \, \mathrm{d}x=\log_e x +c \\ &\int k\, \mathrm{d} x=k x+c \end{aligned}$$

Inverse Trigonometric Integration Chart

Below is the chart for integrals of inverse trigonometric functions:

$$ \begin{aligned} &\int \sin^{-1} x \, \mathrm{d} x=x \sin^{-1} x+\sqrt{1-x^2}+c \\&\int \cos^{-1} x \, \mathrm{d} x=x \cos^{-1} x-\sqrt{1-x^2}+c \\&\int \tan^{-1} x \, \mathrm{d} x=x \tan^{-1} x-\frac{1}{2} \ln \left(x^2+1\right) +c\\&\int \cot^{-1} x \, \mathrm{d} x=x \cot^{-1} x+\frac{1}{2} \ln \left(x^2+1\right) +c\\ &\int \csc ^{-1} x \, \mathrm{d} x=x \csc ^{-1} x+\ln \left|x+\sqrt{x^2-1}\right|+c \\ &\int \sec ^{-1} x \, \mathrm{d} x=x \sec ^{-1} x-\ln \left|x+\sqrt{x^2-1}\right|+c \end{aligned}$$

Other Integration formulas chart

There are some other integration formulas, which are listed below: $$ \begin{aligned} &\int e^{c x} \sin b x \, \mathrm{d} x=\frac{e^{c x}}{c^2+b^2}(c \sin b x-b \cos b x) +c\\ &\int e^{c x} \cos b x \, \mathrm{d} x=\frac{e^{c x}}{c^2+b^2}(c \cos b x+b \sin b x) +c\\&\int \frac{1}{x^2+a^2} \, \mathrm{d} x=\frac{1}{a} \tan^{-1} \frac{x}{a} +c\\&\int \frac{1}{x^2-a^2} \, \mathrm{d} x= \left\{\begin{array}{l} \displaystyle \frac{1}{2 a} \ln \left( \frac{a-x}{a+x} \right) +c\\\displaystyle \frac{1}{2 a} \ln \left( \frac{x-a}{x+a} \right)+c \end{array}\right. \end{aligned}$$

Trigonometric Integration formulas Chart

Let us revisit the trigonometric formulas with some squared formulas and hyperbolic formulas as well:

$$ \begin{aligned} &\int \sin x \, \mathrm{d} x=-\cos x+c \\&\int \cos x \, \mathrm{d} x=\sin x+c \\&\int \tan x \, \mathrm{d} x=\ln |\sec x|+c \\&\int \sec x \, \mathrm{d} x=\ln |\tan x+\sec x|+c \\&\int \sin ^2 x \, \mathrm{d} x=\frac{1}{2}(x-\sin x \cos x)+c \\&\int \cos ^2 x \, \mathrm{d} x=\frac{1}{2}(x+\sin x \cos x)+c \\&\int \tan ^2 x \, \mathrm{d} x=\tan x-x+c \\&\int \sec ^2 x \, \mathrm{d} x=\tan x+c \\ &\int \sin x \cos ^n x \, \mathrm{d} x=-\frac{\cos ^{n+1} x}{n+1}+c \\ &\int \sin ^n x \cos x \, \mathrm{d} x=\frac{\sin ^{n+1} x}{n+1} +c\\&\int \sinh x \, \mathrm{d}x=\cosh x +c \\ &\int \cosh x \, \mathrm{d}x=\sinh x +c\end{aligned} $$

Some of the above formulas are themselves derived from the other ones, we take them as formulas since they appear quite often while solving integrals.

Examples of Integration Formula

Now let us apply these formulas and solve a couple of integrals. We shall use the above formulas without providing their proof, unless we are asked to provide one specifically.