Let n2 be a natural number and 0

Question

Let $n\ge2$ be a natural number and $0<	heta<\dfrac{\pi}2$. Then $\displaystyle\int\frac{(\sin^n	heta-\sin	heta)^{1/n}\cos	heta}{\sin^{n+1}	heta}\,d	heta$ is equal to:

Accepted Answer

Let $\sin\theta=u\Rightarrow\cos\theta\,d\theta=du$:
$$I=\int\frac{(u^n-u)^{1/n}}{u^{n+1}}du=\int u^{-n}(1-u^{1-n})^{1/n}du$$

Let $1-u^{1-n}=v\Rightarrow-(1-n)u^{-n}du=dv\Rightarrow u^{-n}du=\dfrac{dv}{n-1}$:
$$I=\frac1{n-1}\int v^{1/n}dv=\frac1{n-1}\cdot\frac{v^{\frac{n+1}n}}{\frac{n+1}n}+C=\frac{n}{n^2-1}v^{\frac{n+1}n}+C=\frac{n}{n^2-1}\left(1-\frac1{u^{n-1}}\right)^{\frac{n+1}n}+C$$

$$=\frac{n}{n^2-1}\left(1-\frac1{\sin^{n-1}\theta}\right)^{\frac{n+1}n}+C$$

Solutions · 1