What is the integral of x/(tan^2(x)) ?

The integral of x/(tan^2(x)) is -x^2-xcot(x)+(x^2)/2+ln|sin(x)|+C

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integral of x/(tan^2(x))

\int \frac{x}{tan ^{2} ( x )} d x

- x^{2} - x cot (x) + \frac{x ^{2}}{2} + ln ∣ sin (x) ∣ + C

Solution steps

\int \frac{x}{tan ^{2} ( x )} d x

Apply Integration By Parts

: x (- x - cot (x)) - \int - x - cot (x) d x

= x (- x - cot (x)) - \int - x - cot (x) d x

\int - x - cot (x) d x = - \frac{x ^{2}}{2} - ln ∣ sin (x) ∣

= x (- x - cot (x)) - (- \frac{x ^{2}}{2} - ln ∣ sin (x) ∣)

Simplify

x (- x - cot (x)) - (- \frac{x ^{2}}{2} - ln ∣ sin (x) ∣) : - x^{2} - x cot (x) + \frac{x ^{2}}{2} + ln ∣ sin (x) ∣

= - x^{2} - x cot (x) + \frac{x ^{2}}{2} + ln ∣ sin (x) ∣

Add a constant to the solution

= - x^{2} - x cot (x) + \frac{x ^{2}}{2} + ln ∣ sin (x) ∣ + C

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What is the integral of x/(tan^2(x)) ?
The integral of x/(tan^2(x)) is -x^2-xcot(x)+(x^2)/2+ln|sin(x)|+C