Question

\(\displaystyle \int \sin^{3}\theta\ \csc^{2}\theta\,d\theta=\ \ ?\)

• A. \(c+\theta\)
• B. \(c+\cos\theta\)
• C. \(c-\cos\theta\)
• D. \(c+\sin\theta\)

Hint:

Rewrite reciprocals ($\csc^2=1/\sin^2$) and cancel powers first.

Answer 1

The Correct Option is C

\(\csc^{2}\theta=\dfrac{1}{\sin^{2}\theta}\). So \[ \sin^{3}\theta\cdot\csc^{2}\theta=\sin^{3}\theta\cdot\frac{1}{\sin^{2}\theta}=\sin\theta. \] Then \(\int \sin\theta\,d\theta=-\cos\theta+c\).