What is the integral of xsin(x^2)cos(x^2) ?

The integral of xsin(x^2)cos(x^2) is -1/8 cos(2x^2)+C

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integral of xsin(x^2)cos(x^2)

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

- \frac{1}{8} cos (2 x^{2}) + C

Solution steps

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

Rewrite using trig identities

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

Take the constant out:

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

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

Apply u-substitution

= \frac{1}{2} \cdot \int \frac{sin ( u )}{4} d u

Take the constant out:

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

= \frac{1}{2} \cdot \frac{1}{4} \cdot \int sin (u) d u

Use the common integral:

\int sin (u) d u = - cos (u)

= \frac{1}{2} \cdot \frac{1}{4} (- cos (u))

Substitute back

u = 2 x^{2}

= \frac{1}{2} \cdot \frac{1}{4} (- cos (2 x^{2}))

Simplify

\frac{1}{2} \cdot \frac{1}{4} (- cos (2 x^{2})) : - \frac{1}{8} cos (2 x^{2})

= - \frac{1}{8} cos (2 x^{2})

Add a constant to the solution

= - \frac{1}{8} cos (2 x^{2}) + C

What is the integral of xsin(x^2)cos(x^2) ?
The integral of xsin(x^2)cos(x^2) is -1/8 cos(2x^2)+C