Get a free home demo of LearnNext

Available for CBSE, ICSE and State Board syllabus.
Call our LearnNext Expert on 1800 419 1234 (tollfree)
OR submit details below for a call back


Integration by Parts

Have a doubt? Clear it now.
live_help Have a doubt, Ask our Expert Ask Now
format_list_bulleted Take this Lesson Test Start Test

Integration by Parts - Lesson Summary

We try to find the integral of the product of two functions, we can do it in two ways.

∫ f(x).g(x) dx

If one function is the derivative of the other, then we can use the method of substitution.

If that is not the case, then we follow the method of integration by parts.

Suppose u and v are two differentiable functions of a single variable x

Consider ∫ uv dx

d.dx (uv) = u. dv/dx = v. du/dx

∫ d(uv)/dx .dx = ∫ u. dv/dx .dx + ∫ v. du/dx . dx

uv = ∫ u. dv/dx .dx + ∫ v . du/dx . dx

∫ u . dv/dx . dx = uv - ∫ v . du/dx dx

u = f(x) ⇒ du/dx = f '(x)

dv/dx = g(x) ⇒ v = ∫ g(x) dx

∫ f(x) g(x) dx = f(x) ∫ g(x) dx - ∫ [∫ g(x) dx] f '(x) dx

∫ f(x) g(x) dx = f(x) ∫ g(x) dx - ∫ [ f '(x) ∫ g(x) dx]  dx

The integral of the product of two functions = (First function) × (Integral of the second function) - Integral of [(Differential coefficient of the first function) × (Integral of the second function)]

In a given integrand, we often have to judge which function has to be taken as the first function and which one as the second.

This can be easily done with the help of a precedence rule, called ILATE.

Each letter in this acronym stands for a certain kind of function.

'I' stands for inverse functions. Ex: sin-1 x,cos-1 x,....

L for logarithmic functions. EX; log x,....

A for algebraic functions. Ex: ax2+bx+c, x3+3x2+2.....

T for trigonometric functions. EX: sin x , cos x,....

E for exponential functions. EX : ax,ex,....

Ex: ∫ x sin x dx

In this integrand, we have an algebraic function, x and a trigonometric function, sin x.

According to the ILATE rule, x is considered the first function and sin x is considered the second function.

∫ log x . x3 dx

So, we consider the logarithmic function as the first function and the algebraic function as the second function.


Feel the LearnNext Experience on App

Download app, watch sample animated video lessons and get a free trial.

Desktop Download Now
Try LearnNext at home

Get a free home demo. Book an appointment now!