What is Integration? Discuss types of Integration with Example.
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"
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