Integration by Parts | Mathematics (Maths) Class 12 - JEE PDF Download

Definition

dx where u & v are differentiable functions.

Note : While using integration by parts, choose u & v such that
(a)  dx v is simple &  

(b)  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

Remember This:

Proof:

Integrating by parts taking sin bx as the second function,

Again integrating by parts taking cos bx as the second function, we get

Transposing the term -a²/b² I to the left hand side, we get  

Solved Examples

Ex.1 Integrate xⁿ log x

Sol.

Ex.2 Evaluate dx

Sol.

Put sec^–1 x = t so that  

Then the given integral  =  

= t (log t – log e) + c  = sec^–1 x (log (sec^–1 x) – 1) + c  

Ex.3 Evaluate dx.

Sol.

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

Ex.4 Evaluate 

Sol.

We have  

 [ x³ = x(x² + 1) - x]

integrating by parts taking x² as the second function 

Ex.5 Evaluate dx.

Sol.

 (put, 2x + 2 = 3 tanθ ⇒ 2 dx = 3 sec2θ dθ )

Ex.6 If   

Sol.

Ex.7 Evaluate 

Sol.

Ex.8 Evaluate  

Sol.

 x where f(x) = tan x = e^x f(x) + c = e^x tanx + c

Ex.9 Evaluate  

Sol.

using, previous example