Answer to Question #266111 in Calculus for khan

Question

What is Integration? Discuss types of Integration with Example.

Answer 1

The integral of a function $\int_a^b{f(x)dx}$ gives the area under the curve between the limits x=a and x=b.

Integrating refers to the summation of discreet data and is a reverse process of differentiation.

1) integration by parts

This type of integration is applied when integrating between a product of two functions.

Example:

$\int x\ sec^2x\ dx$

$=x\int Sec^2 x\ dx-\int (\int Sec^2x\ dx)(1dx)$

$=x\ tan \ x-\int tan\ x dx$

$=x\ tanx +ln|cos\ x|+C$

2) integration by partial fractions

This type of integration is applied when the function to be integrated are rational proper fraction in the form $\frac{P(u)}{Q(u)}$

Example:

$\int{\frac{2x+3}{(x-3)(x+1)}dx}$

$\frac{2x+3}{(x-3)(x+1)}=\frac{A}{x-3}+\frac{B}{x+1}$

$\implies2x+3=A(x+1)+B(x-3)$

A+B=2

A-3B=3

We solve above to get A=$\frac{9}{4}$ and B=$-\frac{1}{4}$

Now $\int{\frac{2x+3}{(x-3)(x+1)}dx}=\int{\frac{\frac{9}{4}}{x-3}dx}+\int{\frac{-\frac{1}{4}}{x+1}dx}$

$=\frac{9}{4}ln|x-3|-\frac{1}{4}ln|x+1|+C$