Showing all of your working, calculate

Question

Showing all of your working, calculate ∫ t /1 − t^2 dt, using the substitution t = sin θ (and limiting the domain to t ∈ (−1, 1)). Ensure that you write your final answer in terms of t. [Note: this integral can be solved in several ways. What you will do here is not the best way, but is designed to give you practice of the techniques you are learning.

Answer 1

Let t = sin(θ), dt = cos(θ)

When t = -1, θ=3π/2

When t =1, θ=π/2

∫^π/2_3π/2sin(θ)/(1-sin²(θ)) cos(θ)dθ = ∫^π/2_3π/2(sin(θ)/cos²(θ))cos(θ)dθ =∫^π/2_3π/2sin(θ)/cos(θ) dθ

Let u = sin(θ), du=cos(θ) dθ

When θ=3π/2, u = -1

When θ=π/2, u=1

∫¹_-1 u du = u²/2 |¹_-1 = (1-1)/2 = 0

Answer 2