Examples using Methods of Integration
Example 1: Solve [Tex]\bold{\int \log x~dx} [/Tex].
Solution:
Given I = \int \log x~dx = \int \log x.1~dx
The functions are already written according to ILATE rule. Using integration by parts we get,
[Tex]I = \log x\int1dx – \int(\frac{d}{dx}(\log x)\int1dx)dx \\ I = x\log x – \int\frac{1}{x}.x~dx \\ I = x\log x – \int1dx \\ I = x\log x – x + c [/Tex]
Example 2: Solve [Tex]\bold{\int x \sin x~dx} [/Tex]
Solution:
Given I = \int x \sin x~dx
The functions are already written according to ILATE rule. Using integration by parts we get,
[Tex]I = x\int \sin xdx – \int(\frac{d}{dx}x\int\sin x)dx \\ I = -x\cos x- \int-\cos x~dx \\ I = -x \cos x + \int\cos xdx \\ I = -x \cos x + \sin x + c [/Tex]
Example 3: Solve [Tex]\bold{\int (2x^3+1)^7x^2~dx} [/Tex]
Solution:
Given [Tex]I = \int (2x^3+1)^7x^2~dx [/Tex]
Substitute (2x3+1) = u
Differentiate both sides w.r.t x to get
6x2 dx = du
x2dx = du/6
Rewriting the given function as [Tex]I = \int \frac{u^7}{6}du [/Tex]
[Tex]I = \frac{1}{6}(\frac{u^8}{8}) + c \\ I = \frac{u^8}{48} + c \\ I = \frac{(2x^3+1)^8}{48}+c [/Tex]
Example 4: Solve [Tex]\bold{\int \sin^2x ~dx} [/Tex]
Solution:
Given [Tex]I = \int \sin^2x ~dx [/Tex]
We know that cos(2x) = 1 – 2sin2x
(1-cos2x)/2 = sin2x
Substituting the value of sin2x, we get
[Tex]I = \int \frac{(1-\cos(2x))}{2}dx \\ I = \frac{1}{2}[x-\frac{\sin(2x)}{2}] + c \\ I = \frac{x}{2}-\frac{\sin(2x)}{4}+c [/Tex]
Example 5: Solve [Tex]\bold{\int \frac{dx}{x^2-25}} [/Tex].
Solution:
Given
[Tex]I = \int \frac{dx}{x^2-25} = \int \frac{dx}{x^2-5^2} [/Tex]
Using
[Tex]\int \frac{dx}{ (x^2 – a^2)} = \frac{a}{2}log | \frac{x – a} {x + a} | + c \\ I = \frac{5}{2}log | \frac{x – 5} {x + 5} | + c [/Tex]
Methods of Integration
Methods of Integration in Calculus refer to the various techniques that are used to integrate function easily. Many times it is not possible to directly integrate a function, so we need to use a specific technique to reduce the integral and then perform integration. Any method of integration involves identifying the type of integral and then deciding which method to use.
In this article, we will study what is Integration in calculus, methods of integration mainly the method of substitution, Integration by parts, and Integration using Trigonometric Identities.
Table of Content
- What is Integration in Calculus?
- What are Methods of Integration?
- Integration by Parts
- Example of Integration by Parts
- Integration By Substitution
- Example of Integration by Substitution
- Integration using Trigonometric Identities
- Example of Integration using Trigonometric Identities
- Integration by Partial Fraction
- Example of Integration by Partial Fraction
- Integration of Some Special Functions
- Important Points related to Methods of Integration
- Examples using Methods of Integration
- Practice Problems on Methods of Integration