The area bounded by the curves

y = (x + 1)²

y = (x – 1)²

and the line  is

a)

b)

c)

d)

Find the area bounded by the X–axis, part of the curve  and the ordinates at x = 2 and x = 4. If the ordinate x = a divides the area in two equal parts, find a.

a)

b)

c)

d)

Sketch the region bounded by the curves y = x² and  . Find the area.

a)

b)

c)

d)

The value of the integral
 is

a) sin⁻¹ x – 6tan⁻¹(sin⁻¹ x) + c

b) sin⁻¹x – 2(sinx)⁻¹ + c

c) sin⁻¹x – 2(sinx)⁻¹ − 6tan⁻¹(sin⁻¹x) + c

d) sin⁻¹x – 2(sinx)⁻¹ + 5tan⁻¹(sin⁻¹x) + c

Let  for n ≥ 2 and
 

Then equals

a)

b)

c)

d)

Integrate

a)

b)

c)

d)

Find the integral of   

a) tan⁻¹x² + c

b)

c)

d)

Show that  =

 =

a)

b)

c)

d)

 =

a)

b)

c)

d)

 =

a)

b)

c)

d)

 =

a) True

b) False

 =

a)

b)

c)

d)

Show that the integral
 =

Show that
 =   where y =

Show that
 =

Show that   =
 

Show that  =

For any natural number m, show that
 

Show that  =

Let F(x) be an indefinite integral of sin²x
Statement 1: The function F(x) satisfies F(x + π) = F(x) for all real x because
Statement 2: sin²(x + π) = sin²x for all real x

Then which one of the following statements is true?

a) Statement 1 and 2 are true statements and Statement 2 is a correct explanation of Statement 1

b) Statement 1 and 2 are true statements and statement 2 is not a correct explanation of statement 1

c) Statement 1 is true, Statement 2 is false

d) Statement 1 is false, Statement 2 is true

The value of the integral
 is

a)

b)

c) π

d) None of these

Let f : ℝ → ℝ be a differentiable function and f (1) = 4. Then show that the value of   =