Integration By Parts
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
Ex.38 Integrate xn log x
Ex.39 Evaluate dx
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.40 Evaluate dx.
Put x = cos θ so that dx = - sin θ dθ. the given integral
[ x3 = x(x2 + 1) - x]
integrating by parts taking x2 as the second function
Ex.42 Evaluate dx.
(put, 2x + 2 = 3 tanθ ⇒ 2 dx = 3 sec2θ dθ )
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
x where f(x) = tan x = ex f(x) + c = ex tanx + c
using, previous example