Unit - 1

Integral calculus-I

Q1) What is beta function?

A1)

The beta and gamma functions are defined as-

And

These integrals are also known as first and second Eulerian integrals.

Note- Beta function is symmetrical with respect to m and n.

Q2) Evaluate dx

A2)

dx = dx

= γ(5/2)

= γ(3/2+ 1) 

= 3/2 γ(3/2 )  

= 3/2.  ½ γ(½ )

= 3/2. ½ π 

=    ¾ π  

Q3) Find γ(-½) 

A3)

(-½) + 1=  ½

γ(-1/2)  =   γ(-½ + 1)  / (-½)

=   - 2   γ(1/2 )

= - 2 π

Q4) Evaluate  dx.

A4)

Let    dx 

Put or ;dx =2t dt

X	0
t	0

dt

dt

Q5) Evaluate beta function.

A5)

Here we have-

Or

Again integrate by parts, we get-

Repeating the process above, integrating by parts we get-

Or

Q6) What is the relationship between beta and gamma function.

A6)

We know that-

………… (1)

…………………..(2)

Multiply equation (1) by , we get-

Integrate both sides with respect to x within limits x = 0 to x = , we get-

But

By putting λ = 1 + y and n = m + n

We get by using this result in (2)-

So that-

Q7) Evaluate    I   =  

A7)

= 2 π/3

Q8) Prove that-

Where a>0.

A8)

We know that-

Put m – 1  = 1 or m = 2

We get-

Hence proved

Q9) Evaluate

A9)

Let

Put   or  ,,

When,;, 

	o
	1	0

Also

Q10) What is elliptical integral?

A10)

The integral is called elliptic function.

This is derived from the determination of the perimeter of the ellipse.

To evaluate this integral, we first expand the integrand as power series and then integrate term by term.

Elliptic integral of first kind

Elliptic integral of second kind

Here k is modulus and is amplitude.

Q11) Express in terms of elliptic integral.

A11)

Here we have-

Put

Q12) What is error function?

A12)

Error function of x is written as erf(x) and defined as below-

The complementary function of error function of x is written as and defined as below-

Note-

Q13) Find erf(0).

A13)

We know that-

Then

Hence-

Q14) Prove that erf (-x) = - erf (x)

A14)

We know that-

Put t = -u

Q15) Find the mean value of f(t) = over the interval

A15)

As we know that

Here

a = 1 and b = 3 and f(t) =

Q16) What do you understand by Root-mean-square-value of a function?

A16)

If f(t) is defined on the interval , the mean-square value is given by the expression:

This is simply the mean value of over the given interval.

The related quantity: the root-mean-square (r.m.s.) value is given by the following formula.

The r.m.s. Value depends upon the interval chosen. If the values of a or b are changed, then the r.m.s. Value of the function across the interval from a to b will in general change as well. Note that when finding an r.m.s. Value the function must be squared before it is integrated.

Q17) Find the r.m.s value of f(t) = sin t across the interval

A17)

Here a = 0 and b = 2_ so r.m.s can be calculated as,

Note-