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

The integral of (x*sin(x^2)) is -1/2 cos(x^2)+C

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integral of (x*sin(x^2))

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

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

Solution steps

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

Apply u-substitution

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

= \int \frac{sin ( 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 sin (u) d u

Use the common integral:

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

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

Substitute back

u = x^{2}

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

Simplify

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

Add a constant to the solution

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

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