Integration Worksheet4 solution4

In this page integration worksheet4 solution4 we are going to see solution of some practice question from the worksheet of integration.

Question 12

Integrate the following with respect to x, √(1 + cos 2 x)

Solution:

now we are going to use a trigonometric formula for 1 + cos 2 x

(1 + cos 2 x) = 2 cos² x

      = ∫ √(1 + cos 2 x) dx

      = ∫ √(2 cos² x) dx

      = ∫ √2 cos x dx

      = √2 ∫ cos x dx

      = √2 sin x + C


Question 13

Integrate the following with respect to x,  1/(sin²x cos²x)

Solution:

now we are going to use a trigonometric formula for 1

 1= cos² x + sin² x

      = ∫(cos² x+sin² x)/(sin²x cos²x) dx

      = ∫ [cos² x/(sin²x cos²x)] dx + ∫ [sin² x/(sin²x cos²x)] dx

      = ∫ 1/sin²x dx + ∫ 1/cos²x dx

      = ∫ cosec²x dx + ∫ sec²x dx

      = - cot x + tan x + C

      =  tan x - cot x + C


Question 14

Integrate the following with respect to x, sin²x/(1+cos x)

Solution:

now we are going to use a trigonometric formula for sin²x

 sin²x = 1 - cos² x

      = ∫(1 - cos² x)/(1 + cos x) dx

      = ∫ [(1 + cos x) (1- cos x)/(1 + cos x)] dx

      = ∫ (1- cos x) dx

      = ∫ 1 dx - ∫ cos dx

      =  x - (-sin x) + C

      =  x + sin x + C


Question 15

Integrate the following with respect to x, sin 7 x cos 5 x

Solution:

Now we are going to multiply and divide the given question by 2

        = ∫ (1/2) (2sin 7 x cos 5 x) dx

the trigonometric formula for 2 sin A cos B is sin (A + B) + sin (A - B)

        = ∫ (1/2) [sin (7x + 5x) + sin (7x -5x)] dx

        = ∫ (1/2) [sin 12x + sin 2x] dx

        =  (1/2) [- cos 12x/12 - sin 2x/2] + C

        = (-1/2) [cos 12x/12 + sin 2x/2] + C


Question 16

Integrate the following with respect to x, cos 3 x  cos x

Solution:

Now we are going to multiply and divide the given question by 2

        = ∫ (1/2) (2 cos 3 x  cos x) dx

the trigonometric formula for 2 cos A cos B is cos (A + B) + cos (A - B)

        = ∫ (1/2) [cos (3x + x) + cos (3x -x)] dx

        = ∫ (1/2) [cos 4x + cos 2x] dx

        =  (1/2) [sin 4 x/4 + sin 2x/2] + C

integration worksheet4 solution4 integration worksheet4 solution4