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

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

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integral of (xcos^2(x^2))/(sin(x^2))

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

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

Solution steps

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

Rewrite using trig identities

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

Expand

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

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

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

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

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

Simplify

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

Add a constant to the solution

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

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