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

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

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integral of-xsin(x^2)

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

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

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

= - \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