What is the integral of (x*cos(x))/(sin^2(x)) ?

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

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integral of (x*cos(x))/(sin^2(x))

\int \frac{x \cdot cos ( x )}{sin ^{2} ( x )} d x

- x csc (x) + ln tan (\frac{x}{2}) + C

Solution steps

\int \frac{x cos ( x )}{sin ^{2} ( x )} d x

Rewrite using trig identities

= \int \frac{x cot ( x )}{sin ( x )} d x

Rewrite using trig identities

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

Apply Integration By Parts

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

\int - csc (x) d x = - ln tan (\frac{x}{2})

= - x csc (x) - (- ln tan (\frac{x}{2}))

Simplify

= - x csc (x) + ln tan (\frac{x}{2})

Add a constant to the solution

= - x csc (x) + ln tan (\frac{x}{2}) + C

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