# Integral $\int (\cot^{2}x+\cot^{4}x)\mathrm{d}x$

I am having difficulties with integrating this function: $$\int (\cot^{2}x+\cot^{4}x)\mathrm{d}x$$ I separated the function in two and calculated $$\int \cot^2x\mathrm{d}x$$: $$\int \cot^2x\mathrm{d}x=\int (-1+1+\cot^2x)\mathrm{d}x=-\int \mathrm{d}x+\int (1+\cot^2x)\mathrm{d}x=-x-\cot x+c$$ But I don't know how to deal with $$\int \cot^4x\mathrm{d}x$$. I assume that there is a shorter solution somewhere but I'm not sure. In addition, this is a high school level question, I'm only familiar with primitives of simple functions, integration by parts and trigonometric replacement.

Any help would be appreciated, thanks in advance.

Solution $$\int (\cot^{2}x+\cot^{4}x)\mathrm{d}x=\int \cot^{2}x\cdot(\cot^{2}x+1)\mathrm{d}x=\int (\cot^{2}x\cdot\csc^{2}x)\mathrm{d}x$$ Let's assume $$u:=\cot x$$

Then $$\mathrm{d}u=-\csc^2x\mathrm{d}x$$. So we have: $$\int (\cot^{2}x\cdot\csc^{2}x)\mathrm{d}x=-\int u^2\mathrm{d}u=-\frac{u^3}{3}+c=-\frac{\cot^2x}{3}+c$$

N. S. and MyGlasses, thank you for your help.

• At first let $u=\cot x$ – Nosrati Jan 8 '17 at 21:14

$$\int (\cot^{2}x+\cot^{4}x)\mathrm{d}x= \int\cot^{2}x\left(1+\cot^{2}x \right)\mathrm{d}x= \int\cot^{2}x\csc^{2}x \mathrm{d}x$$ and use the fact that $(\cot(x))'=-\csc^2(x)$.
• Well, $1+\cot^2x=\csc^2x$ turns out to be an extremely useful identity. Thank you. – Glycerius Jan 8 '17 at 21:20