In this page integration worksheet5 solution4 we are going to see
solution of some practice question from the worksheet of integration.
Question 11
Integrate the following with respect to x, (1 + tan x)/(x + log sec x)
Solution:
We are going to solve this problem by using substitution method. For that we are going to consider the denominator as "t"
t = (x + log sec x)
dt = 1 + (1/sec x) sec x tan x
dt = 1 + tan x
= ∫ dt/t
= log t + C
= log (x + log sec x) + C
Question 12
Integrate the following with respect to x,e^(m tan⁻¹x)/(1+x²)
Solution:
We are going to solve this problem by using substitution method. For that we are going to consider tan⁻¹x as "t"
t = tan⁻¹x
dt = 1/(1+x²) dx
= ∫e^(m tan⁻¹x)/(1+x²) dx
= ∫e^(m t) dt
= e^(m t)/m + C
= e^(m tan⁻¹x)/m + C
Question 13
Integrate the following with respect to x, x sin⁻¹ (x²)/√(1 - x⁴)
Solution:
We are going to solve this problem by using substitution method. For that we are going to consider tan⁻¹x as "t"
t = sin⁻¹ (x²)
dt = [1/√(1 - (x²)²] (2 x) dx
dt/2 = [1/√(1 - x⁴] x dx
= ∫x sin⁻¹ (x²)/√(1 - x⁴) dx
= ∫ t (dt/2)
= (1/2) ∫t dt
= (1/2) t^(1+1)/(1+1) + C
= (1/2) t²/2 + C
= (1/4) [sin⁻¹ (x²)]² + C
Question 14
Integrate the following with respect to x, 5 (x + 1) (x + log x)⁴/x
Solution:
We are going to solve this problem by using substitution method. For that we are going to consider (x + log x) as "t"
t = (x + log x)
differentiating with respect to "x"
dt = [1 + (1/x)] dx
dt = [(x + 1)/x)] dx
= ∫5 (x + 1) (x + log x)⁴/x dx
= ∫ 5 t ⁴ dt
= 5 t^(4+1)/(4+1) + C
= 5 t⁵/5 + C
= t⁵ + C
= (x + log x)⁵ + C
integration worksheet5 solution4 integration worksheet5 solution4