### Convergence of Improper Integral of First kind

**Definition 1.Â **Â Â If f(x)Â is a bounded and integrableÂ function onÂ , then the improper integral of f(x) onÂ is denoted byÂ and is defined as

provided that the limit on right hand side exists.

The improper integralÂ Â is said to be convergent, ifÂ is eitherÂ , then the integral is said to be divergent.

**Definition 2.Â ** Let f(x) be a bounded integrable function defined onÂ Then the improper integral of f(x) onÂ is denoted byÂ and is defined as

provided that the limit on right hand side exists.

The improper integralÂ Â is said to be convergent, if exists finitely and this limit is called the value of the improper integralÂ is eitherÂ then the integral is said to be divergent.

**Definition 3.Â ** Let f(x) be a bounded integrable function defined onÂ . Then the improper integral of f(x) onÂ is denoted byÂ and is defined as

provided that both the limits on right hand side exist , where c is any real number satisfying k<c<1.

The improper integralÂ is said to be convergent,Â if both the limits on the right- hand side exist finitely and are independent of each other. The improper integralÂ is said to be divergent if the right hand side isÂ

### Example

**(i)**

**Solution:**

Thus exists and is finite. Hence, the given integral is convergent andÂ

**(ii)**Â Â

**Solution:**

So,Â is a divergent integral.

### Convergence of Improper Integral of SecondÂ kind

In this section we shall discussed the convergence of improper integral of second kind.

Recall that a definite integralÂ is an improper integral of second kind if its range of integration is finite and f(x) has at least one point of infinite discontinuity in [a,b]

**Definition 1.Â ** Let f(x) be a bounded function defined on (a,b] such that a is the only point of infinite discontinuity of f(x) i.e.Â . Then the improper integral of f(x) on (a,b] is denoted byÂ and is defined as

provided that the limit on right hand side exists finitely. If l denotes the limit on right hand side, then the improper integralÂ is said to be converge to l, when l is finite. IfÂ Â IfÂ , then the integral is said to beÂ a divergent integral.

**Definition 2.Â ****Â ** Let f(x) be bounded function defined on [a,b) such that b is only point of infinite discontinuity of f(x) i.e.Â Then the improper integral of f(x) on [a,b) is denoted byÂ and is defined as

provided that both theÂ limits on the right hand side exist.

**Definition 3.Â ****Â Â **Let f(x) be bounded function defined on [a,b]-{c} , and c is the only point of infinite discontinuity of f(x) i.e.Â and is defined as

provided that the limit on right hand side exists finitely. The improper integralÂ is said to be convergent if both the limits on the right hand side exists finitely. If either of the two or both the limits on RHS areÂ then the integral is said to be divergent.

### Example

**(i)**

**Solution:**

So, 0 is the only point of infinite discontinuity of f(x) on [0,4)

Hence, the given integral is convergent and its value is 4.

**(ii)**Â

**Solution:**

So, 0 is the only point of discontinuity of f(x) on (0,1].

So, the given integral is divergent.