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sahilgupta_iit

There are four methods of integration:

1) decomposition into sum or difference -------for trigonometric functions

2) integration by parts-------when the expression is a product of 2 terms

3) substitution

4) formulae

i'm postin a few formulae.......

i'm asuuming that u kno the basic formulae

1)when trigonometric functions are given you reduce them to the sum or difference of trigonometric functions....

fro ex: (1- cosx)/(1+cosx) can be reduced to

2cosec²x - 2 cosecx cotx -1 which can b easily integrated

2)INTEGRATION BY PARTS(very important)

u dv = uv -

v du

it's success depends upon the choice of u

(1) if integrand contains any non integrable functions directly from the formula, like tan^-1x, logx we have to take these non integrable functions as u and the other as dv.

(2) if integrand contains both integrable functions, and one of these is xⁿ ( n is positive integer). then we take u=xⁿ .

(3) for other cases choice of u is yours

3) SUBSTITUTION :

x³dx/(1 + x⁴) in this Q we substitute 1+x⁴ = t

so

x³dx = dt-----------------substitute this in the given expression and reduce it in terms of "t"

so we get,

x³dx/(1 + x⁴) = dt/t = log t + c = log(1 + x⁴) + c

4) FORMULAE

certain standard integrals

(a² - x²) dx = (x/2)(

(a² - x²) + (a²/2)sin^-1(x/a) + c

(x² - a²) dx = (x/2)(

(x² - a²) - (a²/2)log[x +

(x² - a²)] + c

(x² + a²) dx = (x/2)(

(x² + a²) + (a²/2)log[x +

(x² + a²)] + c

dx/[

(a² - x²) = sin^-1(x/a) +c

dx/[

(x² - a²) = log[x+

(x² - a²)] + c

dx/[

(x² + a²) = log[x+

(x² + a²)] + c

dx/(a² - x²) = (1/2a)log [ (a+x)/(a-x)] + c

dx/(x² - a²) = (1/2a)log [ (x-a)/(x+a)] + c

dx/(x² + a²) = (1/a)tan^-1(x/a) + c

(e^ax sin bx ) dx = [e^ax/a² + b²][a sinbx - bcosbx] + c

a^xdx = a^x/log_ea + c

f '(x)/f(x) dx = log[f(x)] + c

f ' (x) /

f(x) dx = 2^*

f(x) + c

all these results can be derived using the first three methods

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