5-16 Find the general indefinite integral. 15. \int (1-\sin ^(3) t)/(\sin ^(2) t) d t

Question

5-16 Find the general indefinite integral.

15.
`\int (1-\sin ^(3) t)/(\sin ^(2) t) d t`

Answer 1

Simplify the integrand

`(1-\sin^2(t))/(\sin^2(t)) = (\cos^2(t))/(\sin^2(t)) = \cot^2(t)`

Recall the derivative of csc(t),

`\frac d{dt} \csc(t) = -\cot^2(t)`

It follows that

`\displaystyle \int (1-\sin^2(t))/(\sin^2(t)) \, dt = \int \cot^2(t) \, dt = \boxed{-\csc(t) + C}`