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)

Determine a positive integer n ≤ 5 such that .

a) 1

b) 2

c) 3

d) 4

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 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