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
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 a^{2}/b^{2} I to the left hand side, we get
Ex.1 Integrate x^{n }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
[ x^{3} = x(x^{2} + 1)  x]
integrating by parts taking x^{2} 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
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1. What is integration by parts? 
2. How does integration by parts work? 
3. When should integration by parts be used? 
4. Can integration by parts be used multiple times? 
5. What are some common mistakes to avoid when using integration by parts? 

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