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

The integral of (xcot^2(x^2))/(csc(x^2)) is 1/2 cos(x^2)+1/2 ln|tan((x^2)/2)|+C

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integral of (xcot^2(x^2))/(csc(x^2))

\int \frac{x cot ^{2} ( x ^{2} )}{csc ( x ^{2} )} d x

\frac{1}{2} cos (x^{2}) + \frac{1}{2} ln tan (\frac{x ^{2}}{2}) + C

Solution steps

\int \frac{x cot ^{2} ( x ^{2} )}{csc ( x ^{2} )} d x

Rewrite using trig identities

= \int \frac{x ( - 1 + csc ^{2} ( x ^{2} ) )}{csc ( x ^{2} )} d x

Expand

\frac{x ( - 1 + csc ^{2} ( x ^{2} ) )}{csc ( x ^{2} )} : - \frac{x}{csc ( x ^{2} )} + x csc (x^{2})

= \int - \frac{x}{csc ( x ^{2} )} + x csc (x^{2}) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \int \frac{x}{csc ( x ^{2} )} d x + \int x csc (x^{2}) d x

\int \frac{x}{csc ( x ^{2} )} d x = - \frac{1}{2} cos (x^{2})

\int x csc (x^{2}) d x = \frac{1}{2} ln tan (\frac{x ^{2}}{2})

= - (- \frac{1}{2} cos (x^{2})) + \frac{1}{2} ln tan (\frac{x ^{2}}{2})

Simplify

= \frac{1}{2} cos (x^{2}) + \frac{1}{2} ln tan (\frac{x ^{2}}{2})

Add a constant to the solution

= \frac{1}{2} cos (x^{2}) + \frac{1}{2} ln tan (\frac{x ^{2}}{2}) + C

What is the integral of (xcot^2(x^2))/(csc(x^2)) ?
The integral of (xcot^2(x^2))/(csc(x^2)) is 1/2 cos(x^2)+1/2 ln|tan((x^2)/2)|+C