Integration Worksheet5 solution4

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