Unit – 4

INTEGRAL CALCULUS

Q 1: Find the value of 1/2 ?

Solution:

Let

Then

When

And then t=

Squaring both sides we get

where u and v is any variable in place of t.

   Converting it in polar co-ordinates

  Since 

Q2:

By the definition of gamma function we know that

Putting

Then

When then

then

Similarly

So,  ()(

Converting into polar co-ordinates

Let

Also

Where 

using definition

Or  

Q 3: Compute

     We have  

.

Q 4: Express in terms of gamma function

Given   I = 

Let

Then

Q 5: Prove that

We know that

Putting

….(1)

Again putting

Let

When

Or    ….(2)

From (1) and (2) we get

Q 6: Solve the following

Solution:

= 

=

= (0+1)-

= 1+

=

Q 7: solve the integral

Solution:

Given,

Q 8: Solve the following by using Leibniz Rule

Differentiate 

Solution:

Using,

Q 9: Solve the following

Solution:

And Hence,

To get rid of the constant C , notice that I(1)=0, so

Therefore,