Integration Worksheet Solution4





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

Question 10

(10) Integrate the following with respect to x  1/(1 - cos x)

Solution:

Formula :

(a² - b²) = (a + b) (a - b)

∫ Cos x dx = Sin x + C

                     = ∫ 1/(1 - cos x) dx

Now we are going to multiply by conjugate of (1 - cos x)

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

                     = ∫ (1 - cos x)/[(1² - cos² x)] dx

                     = ∫ (1 - Cos x)/Sin² x dx

                     = ∫ (1/Sin² x) dx - ∫ (Cos x/Sin² x) dx

                     = ∫ Cosec² x dx - ∫ (Cos x/Sin x) x (1/sin x) dx

                     = ∫ Cosec² x dx - ∫ (Cot x) x (Cosec x) dx

                     = - Cot x - (-Cosec x) + C                     

                     = - Cot x + Cosec x + C                     

Answer is - Cot x + Cosec x + C


Question 11

Integrate the following with respect to x    (1- cos 2 x)/(1 + cos 2 x)

Solution:

Formula :

(1 - cos 2x) = 2 sin²x

(1 + cos 2x) = 2 Cos²x

∫ sec² x dx = tan x + C

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

                     = ∫ (2 sin²x)/(2 cos²x) dx

                     = ∫ (Sin²x/cos²x) dx

                     = ∫ tan²x dx

                     = ∫ (sec² x - 1) dx

                     = ∫ (sec² x) dx - dx

                     = tan x - x + C 

Answer is tan x - x + C 


Question 12

Integrate the following with respect to x    (2 tan x - 3 cot x)²

Solution:

Formula :

(a - b)²= a² - 2 a b + b²

∫ Cos x dx = Sin x + C

          = ∫ (2 tan x - 3 cot x)² dx

          = ∫ [ (2 tan x)² - 2 (2 tan x)(3 cot x) + (3 cot x)² ]dx

          = ∫ [ 4 tan² x - 12 tan x cot x + 9 cot² x ]dx

          = ∫ (4 tan² x) dx - ∫ 12 tan x cot x dx + ∫ 9 cot² x dx

          = 4 ∫ (tan² x) dx - ∫ 12 tan x (1/tan x) dx + 9 ∫ cot² x dx

          = 4 ∫ (sec² x - 1) dx - 12 ∫ dx + 9 ∫ (Cosec² x - 1) dx

          = 4 [∫ (sec² x) dx  - ∫ dx] - 12 ∫ dx + 9 [∫ (Cosec² x dx  - ∫ 1 dx]

          = 4 [tan x  - x] - 12 x + 9 [- cot x - x] + C

          = 4 tan x  - 4 x - 12 x - 9 cot x - 9 x + C

          = 4 tan x  - 9 cot x - 9 x - 4 x - 12 x + C

          = 4 tan x  - 9 cot x - 25 x + C

Answer is 4 tan x  - 9 cot x - 25 x + C   integration worksheet solution4

integration worksheet solution4 integration worksheet solution4