Integration: Definition, Properties and Formulas

List of all Integral formulas | Special Integration Formula |
Integration formulas with examples


  Some Basic Integration Formulas 

1.    $\int dx=x+c$
2.    $\int x^n dx=\frac{x^{n+1}}{n+1}+c; (n \neq -1)$
3.    $\int e^x dx=e^x+c$
4.    $\int a^x dx=\frac{a^x}{\log|a|}+c; (a>0$ but $a \neq 1)$
5.    $\int \frac{1}{x} dx=\log x+c$
6.    $\int \frac{dx}{\sqrt{1-x^2}}=\sin^{-1} x+c$
7.    $\int \frac{dx}{1+x^2}=\tan^{-1} x+c$

integration Formulas of Trigonometric Functions

1.    $\int \sin x dx=-\cos x+c$
2.    $\int \cos x dx=\sin x+c$
3.    $\int \sec x dx=\log |\sec x + \tan x|+c$
4.    $\int \text{\co\sec} x dx=\log |\text{\co\sec} x – \cot x|+c$
5.    $\int \tan x dx=\log |\sec x|+c$
6.    $\int \cot x dx=\log |\sin x|+c$
7.    $\int \sec^2 x dx=\tan x+c$
8.    $\int \text{\co\sec}^2 x dx=-\cot x+c$
9.    $\int \sec x \tan x dx=\sec x+c$
10.    $\int \text{cosec} x \cot x dx=-\text{cosec} x+c$

  Some Integration Formulas by Substitution

1.    $\int e^{kx} dx=\frac{e^{kx}}{k}+c$
2.    $\int a^{kx} dx=\frac{a^{kx}}{k\log|a|}+c; (a>0$ but $a ne 1)$
3.    $\int \sin kx dx=-\frac{\cos kx}{k}+c$
4.    $\int \cos kx dx=\frac{\sin kx}{k}+c$
5.    $\int \sec^2 kx dx=\frac{\tan kx}{k}+c$
6.    $\int \text{cosec}^2 kx dx=-\frac{\cot kx}{k}+c$
7.    $\int \sec kx \tan kx dx=\frac{\sec kx}{k}+c$
8.    $\int \text{cosec} kx \cot mx dx=-\frac{\text{cosec} kx}{k}+c$

  Special Functions Integration Formulas 

1.    $\int \frac{dx}{x^2+a^2}=\frac{1}{a}\tan^{-1}\frac{x}{a}+c$
2.    $\int \frac{dx}{x^2-a^2}=\frac{1}{2a}\log |\frac{x-a}{x+a}|+c$
3.    $\int \frac{dx}{a^2-x^2}=\frac{1}{2a}\log |\frac{a-x}{a+x}|+c$
4.    $\int \frac{dx}{\sqrt{x^2+a^2}}=\log |x+\sqrt{x^2+a^2}|+c$
5.    $\int \frac{dx}{\sqrt{x^2-a^2}}=\log |x-\sqrt{x^2-a^2}|+c$
6.    $\int \frac{dx}{\sqrt{a^2-x^2}}=\sin^{-1}\frac{x}{a}+c$
7.    $\int \frac{dx}{x\sqrt{x^2-a^2}}=\frac{1}{a}\sec^{-1}\frac{x}{a}+c$
8.    $\int \sqrt{x^2+a^2} dx$
$=\frac{x\sqrt{x^2+a^2}}{2}+\frac{a^2}{2}\log |x+\sqrt{x^2+a^2}|+c$
9.    $\int \sqrt{x^2-a^2} dx$
$=\frac{x\sqrt{x^2-a^2}}{2}+\frac{a^2}{2}\log |x+\sqrt{x^2-a^2}|+c$
10.    $\int \sqrt{a^2-x^2}dx=\frac{x\sqrt{a^2-x^2}}{2}+\frac{a}{2}\sin^{-1}\frac{x}{a}+c$

  Integration by Parts Formulas 

To find the integration of product of functions, we use the technique of integration by parts. If $u$ and $v$ are two functions of $x$, then the integration formula of their product, that is, the integration of $u v dx$ is given as follows:
 $\int u v dx=u \cdot \int v dx – \int [\frac{du}{dx}( \int v dx )] dx$
The above formula is called the formula of integration by parts. In this formula, we call the function $u$ as the first function and $v$ as the second function. If we have to find the integration of the product of functions, then we have to choose the first or \second function according to the rule below (LIATE rule):
  L -> \logarithmic functions (example $\log x$)
  I ->  inverse functions (example $\sin^{-1}x$)
  A -> algebraic functions (example $x^2$)
  T -> trigonometric functions (example $\sin x$)
  E -> exponential functions (example $e^x$)
For example, if we have to evaluate $\int e^x \sin x dx$ u\sing integration by parts formula, then according to the LIATE rule, we have to take $\sin x$ as the first function and $e^x$ as the \second function.
Some applications of the above formula 
(i)  $\int e^x[f(x)+f'(x)] dx=e^x f(x)+c$
(ii)  $\int e^{ax} \sin bx dx=\frac{e^{ax}(a\sin bx – b \cos bx)}{a^2+b^2}+c$
(iii)  $\int e^{ax} \cos bx dx=\frac{e^{ax}(a\cos bx + b \sin bx)}{a^2+b^2}+c$

  Properties of Definite Integrals

1. If $f(x)=\phi'(x)=\frac{d}{dx}(\phi(x))$ then $\int_a^b f(x) dx=\phi(b)-\phi(a)$
The above is the main property of definite integrals.
2. $\int_a^b f(x) dx=\int_a^b f(t) dt$
3. $\int_a^b f(x) dx=-\int_b^a f(x) dx$
4. $\int_a^b f(x) dx=\int_a^c f(x) dx + \int_b^c f(x) dx$ where $a<c<b.$
5. $\int_a^a f(x) dx=0$
6. $\int_0^a f(x) dx=\int_0^a f(a-x) dx$
7. If $f(a+x)=f(x)$ then  $\int_0^{na} f(x) dx=n \int_0^a f(x) dx$
8. $\int_0^{2a} f(x) dx=2\int_0^a f(x) dx \quad$ if $f(2a-x)=f(x)$
9. $\int_0^{2a} f(x) dx=0 \quad $ if $f(2a-x)=-f(x)$
10. $\int_{-a}^{a} f(x) dx=2\int_0^a f(x) dx \quad$ if $f(-x)=f(x)$
11. $\int_{-a}^{a} f(x) dx=0 \quad$ if $f(-x)=-f(x)$
12. $\int_a^b f(x) dx=\int_a^b(a+b-x) dx$
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