Improper Integrals

IMPROPER INTEGRALS

The definite integral is defined as the limit of a sum under two conditions (i) the interval [a, b] is of finite length and (ii) f is defined and bounded on [a, b].

Then is called a proper integral.

But there are many practical problems where ƒ is unbounded on [a, b] or the interval is not finite. Such integrals are known as improper integrals.

For example: are improper integrals.

1. Kinds of Improper Integrals and their Convergence

(a) Improper integrals of the first kind

Note The integral for every b ≥ a is analogus to "partial sum” in infinite series and so it may be considered as "partial integral".

(b) Improper integrals of the second kind

If the interval [a, b] is finite and f is unbounded at one or more points on [a, b], then

is called an improper integral of the second kind.

1. If f is unbounded at a only (i.e., ƒ has an infinite discontinuity at a), then we define

(ii) the limit exists and is equal to A.

Remark:

Cauchy's Principal Value

WORKED EXAMPLES

Problems based on improper integral of the first kind

Example 1 

Evaluate the improper integral if it exists.

Solution

Example 2 

Evaluated if it exists.

Solution

Example 3 

Evaluate

Solution

When x = a, t = a2 and when x = 0, t = 0

Example 4 

Evaluate

Solution

Example 5 

Evaluate

Solution

But sin a and cos a oscillate finitely between -1 and +1.

Since a cos a → as a → ∞, limit is ∞

Example 6 

Evaluate

Solution

Example 7 

Evaluate if it exists.

Solution

Example 8 

Solution 

Example 9 

Evaluate

Solution

Put x = 0, then 3 = A – C ⇒ C = A - 3 = 2 − 3 = −1

Equating coefficients of x2,

0 = A + B ⇒ B = -A = -2

Example 10 

Evaluate the improper integral

Solution

When v = 0,

Problems based on improper integral of the second kind

Example 11 

Evaluate the improper integral if it exists.

Solution 

Let

Example 12 

Evaluate the improper integral if it exists.

Solution 

Let

⸫ f(x) is unbounded at x = 1.

⸫ the integrand is unbounded when x = 1 and the interval [0, 2] is finite. 

Hence, it is an improper integral of the second kind.

Example 13 

Evaluate the improper integral if it exists.

Solution 

⸫ f(x) is unbounded at x = 0 and x = 2

⸫ the integrand is unbounded and the interval [0, 2] is finite. 

Hence, it is an improper integral of the second kind.

Example 14 

Evaluate the improper integral

Solution 

Let

As x→0 +, loge x → ∞. So, f(x) is unbounded at x = 0. 

⸫ the integrand is unbounded and the interval [-1, 1] is fininte. 

Hence, it is an improper integral of the second kind.

But

Now

Example 15 

Evaluate

Solution 

Let

Hence, it is an improper integral of the second kind

Here

EXERCISE 

Test the convergence of the following improper integrals using definition. Find the value if convergent.

ANSWERS TO EXERCISE 

2. Tests of Convergence of Improper Integrals

As in the case of series of positive terms, we have tests for convergence of improper integrals with positive integrand.

(a) Tests of convergence of improper integrals of the first kind

We state the following theorems without proof.

Theorem 4.2 

Comparison Test

Theorem 4.3 

Note The above result is equivalently, if   diverges, then diverges.

Limit form of comparison test

Theorem 4.4 

That is both the integrals behave alike.

Note 

As in the case of series, here also we consider the following improper integrals for comparison,

Absolute Convergence

WORKED EXAMPLE:

Example 1 

Test the convergence of

Solution 

Example 2 

Test the convergence of

Solution 

⸫ by the comparison test,

Example 3 

Test the convergence of

Solution

Example 4 

Test the convergence of

Solution 

Let

⸫ 0 is not a point of infinite discontinuity.

Example 5 

Test the convergence of

Solution 

Example 6 

Test the convergence of

Solution 

Example 7 

Test the convergence of

Solution 

Example 8 

Test the convergence of

Solution 

(b) Test of convergence of Improper Integrals of the second kind

We state the following theorems without proof.

Theorem 4.5 

Comparison Test

Theorem 4.6 

Limit form of comparison test

Theorem 4.7 

Note

Improper integrals for comparison

WORKED EXAMPLES

Example 1 

Test the convergence of the improper integral

Solution 

Example 2 

Test the convergence of the improper integral

Solution 

Example 3 

Test the convergence and evaluate the improper integral

Solution 

Example 4 

Test the convergence of the improper integral

Solution

Example 5 

Prove that converge.

Solution

The function is bounded, but the interval is infinite.

So, it is improper integral of the first kind.

⸫ 0 is not a discontinuity.

Note Similarly, we can prove is convergent.

These two integrals are called Fresnel's integrals.

They are useful in explaining the concept of light diffraction.

Example 6 

Show that converges.

Solution 

Example 7 

Solution 

Let

Integrating by parts,

EXERCISE

Test the convergence of the following improper integrals.

ANSWERS TO EXERCISE 

1. convergent

2. divergent

3. convergent

4. divergent 

5. convergent

6. convergent

7. convergent

8. divergent

9. convergent

10. convergent

11. convergent 

12. convergent 

13. convergent 

14. convergent 

15. divergent

16. divergent 

17. convergent if n < 2

18. convergent