In this page integration worksheet6 solution6 we are going to see
solution of some practice question from the worksheet of integration.

**Question 14**

Integrate the following with respect to x, tan⁻¹ [(3 x - x³)/(1 - 3 x²)]

**Solution:**

Here we are going to apply the substitution method to integrate this problem

x = tan θ

Now we are going to apply x by tan θ in the given question,So tan⁻¹[(3x-x³)/(1-3x²)] will become

= tan⁻¹[(3tanθ - (tanθ)³)/(1-3(tan θ)²)]

= tan⁻¹[(3tanθ - tan³θ)/(1-3tan²θ)]

= tan⁻¹(tan 3 θ)

= 3 θ

By using the substitution method we have changed the original question as a simple term 3 θ. From this we come to know that integrating the given question is similar of integrating 3 θ. now we are going to replace θ by tan⁻¹ x

y = 3 ∫ tan⁻¹ x dx

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

u = tan⁻¹ x dv = dx

du = 1/(1+x²) v = x

∫ u dv = uv - ∫ v du

= (tan⁻¹ x) x - ∫ x [1/(1+x²)] dx

= (tan⁻¹ x) x - ∫ x/(1+x²) dx

t = 1 + x²

dt = 2 x dx

x dx = dt/2

= (tan⁻¹ x) x - ∫ (dt/2)(1/t)

= (tan⁻¹ x) x - (1/2)∫ 1/t dt

= (tan⁻¹ x) x - (1/2) log t + C

= (tan⁻¹ x) x - (1/2) log (1+x²) + C

**Question 15**

Integrate the following with respect to x, x sin⁻¹ (x²)

**Solution:**

Here we are going to apply the substitution method to integrate this problem

x² = t

2 x dx = dt

x dx = dt/2

= ∫ x sin⁻¹ (x²) dx

= ∫ sin⁻¹ t (dt/2)

= (1/2) ∫ sin⁻¹ t dt

now we are going to apply the substitution method

u = sin⁻¹ t dv = dt

du = 1/√(1-t²) v = t

∫ u dv = u v - ∫ v du

= (1/2) [(sin⁻¹ t)t -∫ t (1/√(1-t²))dt]

= (1/2) [t(sin⁻¹ t) -∫ t/√(1-t²)dt] ------ (1)

now let us find the integration value of ∫ t/√(1-t²) dt after that we can apply the value in the first equation.

let a = 1 - t²

da = - 2 t dt

-da/2 = t dt

∫ t/√(1-t²) dt = ∫ (-da/2)/√a

= - (1/2)∫ (1/√a) da

= - (1/2) [√a/(1/2)]

= - (1/2) [2√a]

= - √(1-t²) + C

now we are going to apply this in the first equation

= (1/2) [x²(sin⁻¹ x²) - (- √(1-t²))]+ C

= (1/2) [x²(sin⁻¹ x²) + √(1-(x²)²)]+ C

= (1/2) [x²(sin⁻¹ x²) + √(1-x⁴)]+ C

integration worksheet6 solution6 integration worksheet6 solution6

- Back to worksheet
- 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