**In this section,Â we shall discuss four integrals of the formÂ when P and Q are polynomial functions of x.**

### Type I

**Integrals of the formÂ where P and Q both are linear function of x**

To evaluate this type of integrals we putÂ i.e., to evaluate integrals of the formÂ Â

The following examples illustrate the procedure.

### Example

**Solution:**

Here P and Q both are linear, so we put

### Type II

**Integrals of the formÂ where P is a quadratic expression and Q is a linear expression**

To evaluate this type of integrals we putÂ i.e. to evaluate integrals of the form

### Example

**Solution:**

Putting y=-1 , 3 respectively in (ii) , we getÂ

Substituting the values of A and B in (i) we obtain

### Type III

**Integrals of the formÂ where P is a linear expression and Q is a quadratic expression**

To evaluate this type of integrals we putÂ i.e. to evaluate integrals of the formÂ

### Example

**Solution:**

### Type IV

**Integrals of the formÂ where P and Q both are pure quadratic expression inÂ x i.e.Â **

To evaluate this type of integrals we putÂ and thenÂ i.e., to evaluate integrals of the form

### Example

**Solution:**