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

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- Integration
- Substitution method
- Decomposition method
- Properties of integrals
- Integration-by parts
- Integration-of Sec³ x
- Standard integrals
- Integrating quadratic denominator
- Integration-using partial fractions
- Definite integrals