Type I:Â 
1. Where m, n belongs to natural number. |
2. If one of them is odd, then substitute for term of even power. |
3. If both are odd, substitute either of them |
4. If both are even, use trigonometric identities only. |
5. If m and n are rational numbers and |
Example
Solution:Â
Type II: 
To evaluate this type of integrals we express as the sum or difference of two squares by using the following algorithm.
1. Make the coefficient of |
2. Add and subtract the square of the half of coefficient of x to express |
3. Use the suitable formula from the following formulas : |
Example
Solution:Â
Type III: Integrals of the type 
Algorithm
1. Make the coefficient of |
2. Find half of the coefficient of x |
3.Add and subtract |
4. Use the suitable formula from the following formulas: |
Example
Solution:
Type IV: Integrals Reducible to the form 
Evaluate:
Solution:
Type V: Integrals of the form 
To evaluate this type of integrals, we use the following algorithm:
Algorithm
1. Write the numerator px+q in the following form: |
2. Obtain the values of |
3. Replace |
4. Integrate RHS in step III and put the values of |
Example
Solution:
Type: VI Integrals of the formÂ
 Â
Where p(x) is a polynomial of degree greater than or equal to 2
To evaluate this type of integrals we divide the numerator by the denominator and express the integral as
Where R(x) is a linear function of x.
Now to evaluate the second integral on RHS apply the method discussed earlier.
Example
Solution:
Type: VII Integrals of the form 
Algorithm
1. Write the numerator px+q in the following form: |
2. Obtain the values of |
3. Replace latex \displaystyle px+q=\lambda \left( 2ax+b \right) \mu &s=1 $ in the given integral to get |
4. Integrate rhS in step III and put the values of |
Example
Solution:
Type: VIII Integrals of the form
Algorithm
1. Divide numerator and denominator both by |
2. Replace |
3. Put This substitution reduces the integral in the form |
Evaluate the integral obtained in step III by using the methods discussed earlier |
Example
Solution:
Dividing numerator and denominator by Â
Type: IX Integrals of the form
To evaluate this type of integrals, we use the following algorithm:
Algorithm
1. Put |
2. Replace |
3. |
4. Evaluate the integral in step 3 by using methods discussed earlier |
Example
Solution:
Put so that
Type: X Alternative method to evaluate integrals of the form 
To evaluate this type of integrals we substituteÂ
Example
Solution:
Type: XI Integrals of the form 
To evaluate this type of integrals, we use the following algorithm:
Algorithm
1. Write Numerator= |
2. Obtain the values of |
3. Replace numerator in the integrand by |
Example
Solution:
Comparing the coefficients of sin x and cos x on both sides, we get
Let t= cos x + sin x then,
Type: XI Integrals of the form  
To evaluate this type of integrals, we use the following algorithm:
Algorithm
1. Write Numerator= |
2. Obtain the values of |
3. Replace the numerator in the integrand by |
4. Evaluate the integral on RHS in step 3 by using the methods discussed earlier |
Example
Solution:
Comparing the coefficients of sin x , cos x and constant term on both we get