Compute  

giving the answer in the form
 

Evaluate

Compute  

Compute  

If a, b are positive real numbers compute  

Compute  

Evaluate  

Evaluate   

Evaluate   

Evaluate   

If α be a real number, compute the integral
 

Evaluate I = where a is a positive constant

Let P(x) be a polynomial with real coefficients. Prove that

Let n ≥ 0 be an integer. Compute the integral  

Compute the integral  

Evaluate  , n ≥ 0

Evaluate   Hence deduce that

Compute the integral  

Let f(x) be a non constant differentiable function defined on (−∞, ∞) such that f(x) = f(1 – x) and  then

a)  vanishes at twice an (0, 1)

b)

c)

d)

  is

Area bounded by  and

Solution of the differential equation is

 is equal to

a)  0

b)  

c)  2

d)  None of these