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,