Question

How do you find the definite integral of `int (1-costheta)/(theta-sintheta)` from `[1,2]`?

Answer 1

`ln(2-sin(2))-ln(1-sin(1))~~1.9286`

Explanation:

Step 1. Use `u`-substitution
Let `u=theta-sin(theta)`
`(du)/(d theta)=(1-cos(theta))d theta`

Step 2. Use this new information to create new limits of integration
`theta=1` becomes `u=1-sin(1)`
`theta=2` becomes `u=2-sin(2)`

Step 3. Plug these substitutions into the integral
`int_(1-sin(1))^(2-sin(2))1/u du`

Step 4. Integrate equation with `u`
`int_(1-sin(1))^(2-sin(2))1/u du=[ln(u)]_(1-sin(1))^(2-sin(2))`
`=ln(2-sin(2))-ln(1-sin(1))~~1.9286`