Integration by Substitution/Replacement
The integration by substitution rule applies if the integrand consists two parts (functions) in which one function is derivative of other (in some form). The two general structure of this form is
or
That is, if a one part is the derivative of another via the chain rule, we let u to denote a likely candidate for the inner part as f(x), then translate the given function so that it is written entirely in terms of u, with no x remaining in the expression.
If we can integrate this new function of u, then the anti-derivative of the original function is obtained by replacing u by the equivalent expression in x.
It can be further classified into three sub categories
- General substitution
- Powers of sine and cosine
- Trigonometric Substitutions
Solved Examples: General substitution
Evaluate Following Integrals
-
Solution 👉 Click Here
Solution
Here,
derivative of x2 is 2x, therefore, we let
Now
or du=2xdx
Since we have exactly 2xdx in the original integral, we can replace it by du, hence, we have
or
or
or
or
- dx
Solution 👉 Click Here
Solution
Here,
Derivative of is 2x, therefore, we let
Now
or du=2xdx
Since we have exactly 2xdx in the original integral, we can replace it by du, hence, we have
or
or
or
- , assuming that a and b are constants, , and n is a positive integer.
Solution 👉 Click Here
Solution
We let u=ax+b so du=adx or dx=. Then
or
or
or
or
- , assuming that a and b are constants and .
Solution 👉 Click Here
Solution
We let
u=ax+b so du=adx or dx=
Then
or
or
or
Evaluate the following Intrgral
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Solved Examples: Powers of sine and cosine
Evaluate the following integrals
- .
Solution 👉 Click Here
Solution
Here, integrand consists products of the sine, sine with odd exponent, so it can be integrated by using substitution of trigonometric identities. Thus,
or
or
Now, we use
then
Hence, we have
or
or
or
or
or
- .
Solution 👉 Click Here
Solution
Here,integrand consists products of the sine and cosine, sine with even exponent, so it can be integrated by using substitution of trigonometric identities. Thus,
or
or
or
-
Solution 👉 Click Here
Solution
यस प्रश्नमा मानौ , then
Now, the integral is
or
or
or
or
or
-
Solution 👉 Click Here
Solution
यस प्रश्नमा मानौ ,then
Now, the integral is
or
or
or
or
or
or
or
-
Solution 👉 Click Here
Solution
यस प्रश्नमा मानौ , then
Now, the integral is
or
or
or
or
or
or
Trigonometric Substitutions
This type of substitution is usually indicated when the function of integrate contains a polynomial expression to use the fundamental trigonometric identity as
or
or
or
Therefore
- if it contains , we substitute
- if it contains we substitute and
- if it contains , we substitute .
Sometimes, we need something a bit different to handle constants other than one. Thus, key points are as follows
- Put for integration involving
- Put for integration involving
- Put for integration involving
Solved Examples: Trigonometric Substitutions
Evaluate following integrals
-
Solution 👉 Click Here
Solution
This question is in the form , so we substitute, , then , thus we have
or
or
or
or
(A)
Here
or
(1)
Next
or
or
(2)
Thus, using (1) and (2),the solution (A) is
or
-
Solution 👉 Click Here
The question is in the form of , so we let
Then
or
Hence the integral is
or
or
or
or
or
or
or
or
or
or(A)
Here
or
(1)
Next
or
or
or
or
(2)
Thus, using (1) and (2),the solution (A) is
or
or
-
Solution 👉 Click Here
Solution
The question is in the form of , so we let
Then
Hence the integral is
or
or
or
or
or
or
or
or
or(A)
Here
or
(1)
Next
or
or
or
or
(2)
Thus, using (1) and (2),the solution (A) is
or
or
-
Solution 👉 Click Here
Solution
यस प्रश्नमा
मानौ , then
Now, the integral is
or
or
or
Now, सुत्रको प्रयोग गरौ ।
Then
or
or
Evaluate the following Intrgral
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