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