Answer
$$\int \frac{\csc^3\sqrt\theta}{\sqrt\theta}d\theta=-\csc \sqrt\theta\cot \sqrt\theta-\ln|\csc \sqrt\theta+\cot\sqrt\theta|+C$$
Work Step by Step
$$A=\int \frac{\csc^3\sqrt\theta}{\sqrt\theta}d\theta$$
Set $u=\sqrt\theta$, which means $$du=\frac{1}{2\sqrt\theta}d\theta$$ $$\frac{d\theta}{\sqrt\theta}=2du$$
Therefore, $$A=2\int\csc^3udu$$
Use Reduction Formula 100, which states that
$$\int \csc^naxdx=-\frac{\csc^{n-2}ax\cot ax}{a(n-1)}+\frac{n-2}{n-1}\int\csc^{n-2}axdx$$
for $n=3$ and $a=1$:
$$A=2\Big(-\frac{\csc u\cot u}{2}+\frac{1}{2}\int\csc udu\Big)$$ $$A=-\csc u\cot u+\int\csc udu$$ $$A=-\csc u\cot u-\ln|\csc u+\cot u|+C$$ $$A=-\csc \sqrt\theta\cot \sqrt\theta-\ln|\csc \sqrt\theta+\cot\sqrt\theta|+C$$
