Integration Worksheet6 solution2

In this page integration worksheet6 solution2 we are going to see solution of some practice question from the worksheet of integration.

Question 5

Integrate the following with respect to x,   tan⁻ ¹ x

Solution:

Here we are going to use the method partial differentiation to integrate the given question.

∫ tan⁻ ¹ x dx

∫ u dv = u v - ∫ v du

u = tan⁻ ¹ x             dv = dx

du = 1/(1 + x²)         v = x

     = ∫ tan⁻ ¹ x dx

     =  (tan⁻ ¹ x) x - ∫ x [1/(1 + x²)] dx

1 + x² = t

2 x dx = dt

   x dx = dt/2 

     =  (tan⁻ ¹ x) x - ∫ (dt/2)  [1/t] dx

     =  (tan⁻ ¹ x) x - ∫ x [1/(1 + x²)] dx

     =  (tan⁻ ¹ x) x - ∫ x/(1 + x²) dx

now we are going to apply the substitution method to integrate this

t = 1 + x²

dt = 2 x dx

x dx = dt/2

     =  (tan⁻ ¹ x) x - (1/2)∫ dt/t

     =  x (tan⁻ ¹ x) - (1/2) log t + C

     =  x tan⁻ ¹ x - (1/2) log (1 + x²) + C


Question 6

Integrate the following with respect to x,   x tan ² x

Solution:

Here we are going to use the method partial differentiation to integrate the given question.

∫ x tan ² x dx

now we are going to apply trigonometric formula for tan ² x

sec² x - 1 = tan ² x

∫ x tan ² x dx  = ∫ x (sec² x - 1) dx

                    = ∫ (x sec² x - x) dx

                    = ∫ x sec² x dx - ∫ x dx

u = x          dv = sec² x

du = dx        v = tan x

                    = x (tan x) - ∫ tan x dx - ∫ x dx

                    = x (tan x) - ∫ (sin x/cos x) dx - ∫ x dx

                    = x (tan x) - log (cos x) - (x²/2) + C

integration worksheet6 solution2 integration worksheet6 solution2