integration substitution

substitution method for integration

Integration formulae

If an integral is not of a standard type, but can be reduced to a standard type by substitution

of the form
u = f(x)
du = f ' (x) dx take the differential on both sides if necessary

after substitution, remember that everything should be in terms of u alone

Examples on integration by substitution


* integral of x^3 / sqrt(9 - x^2)
answer and some explanation

* integral of (x^3)*sqrt[4-x^2]
answer and some explanation


* integral of 1 / { x*sqrt[ax-x^2]} using substitution
explanation of the answer to integration by substitution

*integral of 1 / 6{ [ x^(1/2)+ x^(1/3) ] } using substitution
answer with some explanation 


*integral of sqrt{[1-sqrt(x)] / [1+sqrt(x)]}
explanation and answer to the problem on integration

*integral of 2*cube of (tanx) with 0 to pi/4 as limits
explanation and answer to integral of 2*cube of (tanx) with 0 to pi/4 as limits




PAGE 1 BASIC INTEGRATION

PAGE 2 INTEGRATION BY SUBSTITUTION
 
PAGE 3 INTEGRATION BY COMPLETION OF SQUARES

PAGE 4 INTEGRATION BY PARTS

PAGE 5 INTEGRATION BY MANIPULATION OF NUMERATOR IN TERMS OF DENOMINATOR

integration guide (certain topics)

free online guide to certain topics on integration for plus two, isc , cbse etc

First of all try to familiarize yourself with the formulae on integration and trigonometric identities Basic Integration
you can integrate an expression term by term
that is you can break up an integral of sum or difference of two expressions into sum or difference of two integrals
but do not do that with a product or quotient of two functionsbut integral of 2f(x) can be expressed as 2 * integral of f(x) since "2" is a constantDefinite Integrals (Integrals with limits)
trying to write expressions in the form x^n
for eg.
cube root of x can be written as x^(1/3)
1 / (x ³) can be written as x^(-3)

if you know that the integral of f(x) is g(x) + C
then the integral of f(ax+b) will be ( 1/a ) g(ax+b) + C

you can divide the numerator term by term using the denominator but not the other way.if the numerator is of higher degree than the denominator use long division

if you have a product of sine and/or cosine terms use trigonometric identities for sinAcosB etc to break them into sum or difference before integrating
you can use substitution methods or trigonometric identities to evaluate integrals with powers of sine or cosines.



Examples on basic integration

integral of sin(5x)sin(8x)
answer and some explanation



 integral of (x^3 +3x +4) /sqrt(x)
answer and some explanation 

integral of [ x^3 - x^2 + x -1] / (x-1)
answer and some explanation

*integral of e^(2-3x) using limit of sums
answer of integral of e^(2-3x) using limit of sums

PAGE 1 BASIC INTEGRATION
PAGE 2 INTEGRATION BY SUBSTITUTION

 PAGE 3 INTEGRATION BY COMPLETION OF SQUARES
PAGE 4 INTEGRATION BY PARTS
PAGE 5 INTEGRATION BY MANIPULATION OF NUMERATOR IN TERMS OF DENOMINATOR

PAGE 6 INTEGRATION USING PARTIAL FRACTIONS
PAGE 7 INTEGRATION OF 1 / [ACOSX +BSINX +C]



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