f(x) = dx^6x is a polynomial of degree

Question

$f(x) = \displaystyle\int \dfrac{dx}{\sin^6x}$ is a polynomial of degree

Accepted Answer

$f(x) = \int \cosec^6x\,dx$

Using the reduction formula $I_n = \int \cosec^nx\,dx = -\dfrac{\cosec^{n-2}x\cot x}{n-1}+\dfrac{n-2}{n-1}I_{n-2}$:

$f(x) = -\dfrac{\cosec^4x\cot x}{5}+\dfrac45\left(-\dfrac{\cosec^2x\cot x}{3}+\dfrac23 I_2\right)$

$= -\dfrac{\cosec^4x\cot x}{5}-\dfrac{4}{15}\cosec^2x\cot x+\dfrac{8}{15}(-\cot x)$

Expressing $\cosec^2x = 1+\cot^2x$ and simplifying fully:

$f(x) = -\dfrac{\cot^5x}{5}-\dfrac23\cot^3x-\cot x + C$

It is a polynomial of degree 5 in $\cot x$.

Solutions · 1