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

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

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

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

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

Solution steps

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

Use the basic trigonometric identity:

\frac{cos ( x )}{sin ( x )} = cot (x)

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

Apply u-substitution

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

= \int \frac{cot ( 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) d u

Rewrite using trig identities

= \frac{1}{2} \cdot \int \frac{cos ( u )}{sin ( u )} d u

Apply u-substitution

: \int \frac{1}{v} d v

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

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= \frac{1}{2} ln ∣ v ∣

Substitute back

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

Add a constant to the solution

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

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