integration of some products using by parts

Integration by parts

Integration formulae

If an integral is a product of the form uv

usually u is chosen so that successive derivatives of u will become zero after a finite number of steps
provided v has a nice integral or you can make the choice using ILATE rule
inverse , log, algebraic, trigonometric, exponential

for eg. for integral of [x² cosx]

u = x² , v = cosx

but for integral of [x² ln(x)] , you should make it into integral of [ ln(x) * x²]
so that u = ln(x) , v = x²

for integral of lnx, make it into integral of [lnx * 1] with u = lnx , v = 1

in some integrals, like integral of [ (e^x) cosx ], take the integral as I, apply integration by parts a couple of times or so, and then obtain the original integral I again, then rearrange and extract the value of I.

also note the type
here split into two integrals
use integration by parts once on integral of [f(x)exp(x)] taking f(x) as the first function
this will cancel off the integral of [f '(x)exp(x)] leaving you with exp(x) * f(x ) + C which is the required answer

Examples on integration by parts


* integral of ln(x)
answer and some explanation


*integral of arc(cosx)
answer and some explanation


* integral of { [ arcsin(sqrt(x)) - arccos(sqrt(x)) ] / [ arcsin(sqrt(x)) + arccos(sqrt(x))] }
explanation and answer to the integral using integration by parts is available here

*integral of [1 / (x^4)][sqrt( 1 + (x^2))][ log( 1 + (x^2)) - 2log( x)]
explanation and answer using substitution and then integration by parts is available here

*integral of  [e^2x]cos3x using integration by parts
explanation and answer of   integral of [e^2x]cos3x using integration by parts


PAGE 1 BASIC INTEGRATION
PAGE 2 INTEGRATION BY SUBSTITUTIONPAGE 3 INTEGRATION BY COMPLETION OF SQUARES
PAGE 4 INTEGRATION BY PARTS
PAGE 5 INTEGRATION BY MANIPULATION OF NUMERATOR IN TERMS OF DENOMINATOR








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