### Type I:

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 is a negative integer, then substitute cot x=p or tan x=p which so ever is found suitable |

### Example

**Solution: **

### 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 unity, if it is not, by multiplying and dividing by it. |

2. Add and subtract the square of the half of coefficient of x to express in the form |

3. Use the suitable formula from the following formulas : |

### Example

**Solution: **

### Type III: Integrals of the type

### Algorithm

1. Make the coefficient of unity, if it is not |

2. Find half of the coefficient of x |

3.Add and subtract inside the square root to express the quantity inside the square root in the form |

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 and by equating the coefficients of like powers of x on both sides. |

3. Replace in the given integral to get: |

4. Integrate RHS in step III and put the values of and obtained in step II. |

### Example

**Solution:**

### Type: VI Integrals of the form

**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 and by equating the coefficients of like powers of x on both sides |

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 and obtained in step II. |

### Example

**Solution:**

### Type: VIII Integrals of the form

#### Algorithm

1. Divide numerator and denominator both by |

2. Replace , if any, in denominator by |

3. Put so that 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

### Type: IX Integrals of the form

To evaluate this type of integrals, we use the following algorithm:

#### Algorithm

1. Put |

2. Replace in the numerator by |

3. so that This substitution reduces the integral form |

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 substitute

### Example

**Solution:**

### Type: XI Integrals of the form

To evaluate this type of integrals, we use the following algorithm:

#### Algorithm

1. Write Numerator= (Diff. of denominator) + (Denominator) |

2. Obtain the values of and by equating the coefficients of |

3. Replace numerator in the integrand by to obtain |

### 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= (Diff. of denominator) + (Denominator) + v |

2. Obtain the values of and by equating the coefficients of and the constant term both sides |

3. Replace the numerator in the integrand by to obtain |

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