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Integration by Substitution

The studying process about the integration by substitution method involves the solving process of the integral function. If we have the function that are not activate or convenient to integrate directly are solved by different methods of the integration. Integration by substitution, also known as u-substitution or change of variables.

Integration by Substitution Method:
Let A : [c, d] -> R be continuously differentiable. Assume A[c, d] = [m, n] with A(c) = m and A(d) = n. If g : [m, n] -> R is continuous, then g(A)A' is Riemann integrable on [c, d] and $\int_c^d$ g(A(x))A'(x) dx = $\int_m^n$g

 Related Calculators Calculator for Substitution Method Calculate Integral Calculate Definite Integrals Calculate Double Integral

Integration by Substitution Examples

Below are the examples on integration by substitution -
Example 1 : Integrate the function  int 10/sqrt(1 - 100x^2) dx.

Solution:

Let I = int 10/sqrt(1 - 100x^2) dx

The step is to be involved for substitution by splitting the term 10/sqrt(1- 100x^2)

Put 10 x = sin u                                    … (1)

=> u = sin?1 (10x)

and 10 dx = cos u du                                   … (2)

? I =  int 10/sqrt(1- 100x^2) dx

=int 1/sqrt(1 - sin^2 u) (cos u du)    by using (1) and (2)

= int 1/sqrt(cos^2 u) (cos u du)

= int du
= u + c                 (Put u = sin?1(10x) in the function for the variable we assigned)

=>  int 10/sqrt(1 - 100x^2) dx = sin?1 (10x) + c, where c is the constant of integration.

Example 2 :   Integrate the function  int 5/(1 + x^2) dx using the substitution methods.

Solution:

Let I = int 5/(1 + x^2) dx

Put x = tan u                                   … (1)

=>  u = tan?1 x

and dx = sec2 u du                                   … (2)

? I =  int 5/(1+ tan^2u) sec^2u du          by using (1) and (2)

=   int 5/(sec^2u) sec^2u du

= int 5du

= 5u + c                 (Put  u = tan?1 x in the function for the variable we assigned)

=> int 5/(1+ x^2) dx =  5 tan?1 x  + c

This is the solution obtained after substituting the terms.

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