Integration By Parts Examples
Example 1: Find β« ex x dx.
Solution:
Let I = β« ex x dx
Choosing u and v using ILATE rule
u = x
v = exDifferentiating u
u'(x) = d(u)/dx
β u'(x) = d(x)/dx
β u'(x) = 1
β«v dx = β«ex dx = ex
Using the Integration by part formula,
β I = β« ex x dx
β I = x β«ex dx β β«1 (β« ex dx) dx
β I = xex β ex + C
β I = ex(x β 1) + C
Example 2: Calculate β« x sin x dx.
Solution:
Let I = β« x sin x dx
Choosing u and v using ILATE rule
u = x
v = sin xDifferentiating u
u'(x) = d(u)/dx
β u'(x) = d(x)/dx
β u'(x) = 1
Using the Integration by part formula,
β I = β« x sin x dx
β I = x β«sin x dx β β«1 β«(sin x dx) dx
β I = β x cos x β β«βcos x dx
β I = β x cos x + sin x + C
Example 3: Find β« sinβ1 x dx.
Solution:
Let I= β« sinβ1 x dx
β I = β« 1.sinβ1 x dx
Choosing u and v using ILATE rule
u = sinβ1 x
v = 1Differentiating u
u'(x) = d(u)/dx
β u'(x) = d(sinβ1 x )/dx
β u'(x) = 1/β(1 β x 2)
Using the Integration by part formula,
β I = β« sinβ1 x dx
β I = sinβ1 x β« 1 dx β β« 1/β(1 β x 2) β«(1 dx) dx
β I = x sinβ1 x β β«( x/β(1 β x 2 ) )dx
Let, t = 1 β x 2
Differentiating both sides
dt = β2x dx
β βdt/2 = x dx
β I = β« sinβ1 x dx = x sinβ1 x β β«β(1/2βt ) dt
β I = x sinβ1 x + 1/2β«tβ1/2 dt
β I = x sinβ1 x + t1/2 + C
β I = x sinβ1 x + β(1 β x2 ) + C
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Integration by Parts
Integration by Parts: Integration by parts is a technique used in calculus to find the integral of the product of two functions. Itβs essentially a reversal of the product rule for differentiation.
Integrating a function is not always easy sometimes we have to integrate a function that is the multiple of two or more functions in this case if we have to find the integration we have to use integration by part concept, which uses two products of two functions and tells us how to find their integration.
Now letβs learn about Integration by parts, its formula, derivation, and others in detail in this article.
Table of Content
- What is Integration by Parts?
- What is Partial Integration?
- Integration By Parts Formula
- Derivation of Integration By Parts Formula
- ILATE Rule
- How to Find Integration by Part?
- Repeated Integration by Parts
- Applications of Integration by Parts
- Integration by Parts Formulas
- Integration By Parts Examples
- Practice Problems
- FAQs
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