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5-16 Find the general indefinite integral.

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

User Antnewbee
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1 Answer

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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}

User Amir Mohsen
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