How do you integrate $\int {csc}^{2} (\frac{x}{2}) d x$ $int csc^2(x/2) dx$ ?

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Answer 1 · 2016-12-14T08:55:07

$\int {csc}^{2} (\frac{x}{2}) d x = - 2 cot (\frac{x}{2}) + C$ $" "intcsc^2(x/2)dx=-2cot(x/2)+C$

$\frac{d}{d x} (cot x) = - {csc}^{2} x$ $d/(dx)(cotx)=-csc^2x$

so $\frac{d}{d x} (cot (\frac{x}{2})) = - \frac{1}{2} {csc}^{2} x$ $" "d/(dx)(cot(x/2))=-1/2csc^2x$

inspecting $\int {csc}^{2} (\frac{x}{2}) d x$ $" "intcsc^2(x/2)dx$

we have to adjust for the $- \frac{1}{2}$ $" "-1/2$

$\int {csc}^{2} (\frac{x}{2}) d x = - 2 cot (\frac{x}{2}) + C$ $" "intcsc^2(x/2)dx=-2cot(x/2)+C$

How do you integrate int csc^2(x/2) dx∫csc2(x2)dxint csc^2(x/2) dx?