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

The integral of (xcos(x^2))/(sin^2(x^2)) is -1/2 csc(x^2)+C

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

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

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

Solution steps

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

Rewrite using trig identities

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

Use the basic trigonometric identity:

\frac{1}{sin ( x )} = csc (x)

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

Apply u-substitution

: \int \frac{cot ( u ) csc ( u )}{2} d u

= \int \frac{cot ( u ) csc ( u )}{2} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int cot (u) csc (u) d u

Apply u-substitution

: \int - 1 d v

= \frac{1}{2} \cdot \int - 1 d v

Integral of a constant:

\int a d x = a x

= \frac{1}{2} (- 1) v

Substitute back

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

Simplify

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

Add a constant to the solution

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

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What is the integral of (xcos(x^2))/(sin^2(x^2)) ?
The integral of (xcos(x^2))/(sin^2(x^2)) is -1/2 csc(x^2)+C