Show that

Find the value of

a)

b)

c)

d)

Evaluate

a)

b)

c)

d)

Find the area bounded by the X - axis, part of the curve  and the ordinate at x = 2 and x = 4. If the ordinate at x = a divide the area into two equal parts, find a,

a)

b)

c)

d)

Evaluate

a)

b)

c)

d)

Given a function f (x) such that
i) it is integrable over every interval on the real axis and
ii) f (t + x) = f (x) for every x and a real t, then show that the integral  is independent of a.

Evaluate

a)

b)

c)

d)

Evaluate

a)

b)

c)

d)

Evaluate

a)

b)

c)

d)

If f (x) and g (x) are continuous functions on (0, a) satisfying f (x) = f (a – x) and g (x) + g (a – x) = 2 then show that

Show that

Evaluate

a)

b)

c)

d)

Evaluate

Determine the value of

a)

b)

c)

d)

Evaluate

a) πln2

b)

c)

d)

Prove that  Hence or otherwise evaluate the integral .

Evaluate

a)

b)

c)

d)

For x > 0, let  find the function  and show that . Here .

a)

b)

c)

d)

If f (x) is an even function then prove that
 .

Evaluate

The value of the integral  is equal to a

a) True

b) False

The integral  dx where [ ] denotes the greatest integer function equals .  .   .

a)

b)  + 1

c)

d)

The value of  is

a) 0

b) 1

c) 2

d) 4