Integration by Parts - Indefinite Integration JEE Notes | EduRev

Mathematics (Maths) Class 12

JEE : Integration by Parts - Indefinite Integration JEE Notes | EduRev

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Integration By Parts        
Integration by Parts - Indefinite Integration JEE Notes | EduRevdx where u & v are differentiable functions.

Note : While using integration by parts, choose u & v such that
(a) Integration by Parts - Indefinite Integration JEE Notes | EduRev dx v is simple &  

(b) Integration by Parts - Indefinite Integration JEE Notes | EduRev dx is simple to integrate.

This is generally obtained, by keeping the order of u & v as per the order of the letter in ILATE, where

I – Inverse function
 L – Logarithmic function
 A – Algebraic function
 T – Trigonometric function
 E – Exponential function

Ex.38 Integrate xlog x

Sol.

Integration by Parts - Indefinite Integration JEE Notes | EduRev

Integration by Parts - Indefinite Integration JEE Notes | EduRev

Integration by Parts - Indefinite Integration JEE Notes | EduRev

Integration by Parts - Indefinite Integration JEE Notes | EduRev

Ex.39 Evaluate Integration by Parts - Indefinite Integration JEE Notes | EduRevdx

Sol.

Put sec–1 x = t so that  Integration by Parts - Indefinite Integration JEE Notes | EduRev

Then the given integral  =  Integration by Parts - Indefinite Integration JEE Notes | EduRev

Integration by Parts - Indefinite Integration JEE Notes | EduRev

= t (log t – log e) + c  = sec–1 x (log (sec–1 x) – 1) + c   Integration by Parts - Indefinite Integration JEE Notes | EduRev

 

Ex.40 Evaluate Integration by Parts - Indefinite Integration JEE Notes | EduRevdx.

Sol.

Put x = cos θ so that dx = - sin θ dθ. the given integral 

Integration by Parts - Indefinite Integration JEE Notes | EduRev

Integration by Parts - Indefinite Integration JEE Notes | EduRev

Integration by Parts - Indefinite Integration JEE Notes | EduRev

Integration by Parts - Indefinite Integration JEE Notes | EduRev

Integration by Parts - Indefinite Integration JEE Notes | EduRev


Ex.41 Evaluate Integration by Parts - Indefinite Integration JEE Notes | EduRev

Sol.

We have  Integration by Parts - Indefinite Integration JEE Notes | EduRev

Integration by Parts - Indefinite Integration JEE Notes | EduRev [ x3 = x(x2 + 1) - x]

integrating by parts taking x2 as the second function 

Integration by Parts - Indefinite Integration JEE Notes | EduRev

Integration by Parts - Indefinite Integration JEE Notes | EduRev

Ex.42 Evaluate Integration by Parts - Indefinite Integration JEE Notes | EduRevdx.


Sol.

  Integration by Parts - Indefinite Integration JEE Notes | EduRev

Integration by Parts - Indefinite Integration JEE Notes | EduRev (put, 2x + 2 = 3 tanθ ⇒ 2 dx = 3 sec2θ dθ )

Integration by Parts - Indefinite Integration JEE Notes | EduRev

Integration by Parts - Indefinite Integration JEE Notes | EduRev

 

Ex.43 If   Integration by Parts - Indefinite Integration JEE Notes | EduRev

Sol.

Integration by Parts - Indefinite Integration JEE Notes | EduRev

Integration by Parts - Indefinite Integration JEE Notes | EduRev

Integration by Parts - Indefinite Integration JEE Notes | EduRev

Integration by Parts - Indefinite Integration JEE Notes | EduRev

Integration by Parts - Indefinite Integration JEE Notes | EduRev

Integration by Parts - Indefinite Integration JEE Notes | EduRev

REMEMBER THIS
Integration by Parts - Indefinite Integration JEE Notes | EduRev

Integrating by parts taking sin bx as the second function,

Integration by Parts - Indefinite Integration JEE Notes | EduRev

Again integrating by parts taking cos bx as the second function, we get
Integration by Parts - Indefinite Integration JEE Notes | EduRev

Integration by Parts - Indefinite Integration JEE Notes | EduRev

Integration by Parts - Indefinite Integration JEE Notes | EduRev

Transposing the term -a2/b2 I to the left hand side, we get  Integration by Parts - Indefinite Integration JEE Notes | EduRev

Integration by Parts - Indefinite Integration JEE Notes | EduRev

Integration by Parts - Indefinite Integration JEE Notes | EduRev

Integration by Parts - Indefinite Integration JEE Notes | EduRev

Integration by Parts - Indefinite Integration JEE Notes | EduRev

Integration by Parts - Indefinite Integration JEE Notes | EduRev

Ex.44 Evaluate Integration by Parts - Indefinite Integration JEE Notes | EduRev


Sol.

Integration by Parts - Indefinite Integration JEE Notes | EduRev

Integration by Parts - Indefinite Integration JEE Notes | EduRev

Integration by Parts - Indefinite Integration JEE Notes | EduRev

Integration by Parts - Indefinite Integration JEE Notes | EduRev

Integration by Parts - Indefinite Integration JEE Notes | EduRev

Integration by Parts - Indefinite Integration JEE Notes | EduRev

Integration by Parts - Indefinite Integration JEE Notes | EduRev

Integration by Parts - Indefinite Integration JEE Notes | EduRev

Integration by Parts - Indefinite Integration JEE Notes | EduRev

Ex.45 Evaluate  Integration by Parts - Indefinite Integration JEE Notes | EduRev

Sol.

Integration by Parts - Indefinite Integration JEE Notes | EduRev

Integration by Parts - Indefinite Integration JEE Notes | EduRev

Integration by Parts - Indefinite Integration JEE Notes | EduRev

Integration by Parts - Indefinite Integration JEE Notes | EduRev x where f(x) = tan x = ex f(x) + c = ex tanx + c

 

Ex.46 Evaluate  Integration by Parts - Indefinite Integration JEE Notes | EduRev

Sol.

Integration by Parts - Indefinite Integration JEE Notes | EduRev

Integration by Parts - Indefinite Integration JEE Notes | EduRev

Integration by Parts - Indefinite Integration JEE Notes | EduRev

Integration by Parts - Indefinite Integration JEE Notes | EduRev

using, previous example Integration by Parts - Indefinite Integration JEE Notes | EduRev

Integration by Parts - Indefinite Integration JEE Notes | EduRev

Integration by Parts - Indefinite Integration JEE Notes | EduRev

 

 

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