What is the integral of xsin(pix^2) ?

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

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integral of xsin(pix^2)

\int x sin (π x^{2}) d x

- \frac{1}{2 π} cos (π x^{2}) + C

Solution steps

\int x sin (π x^{2}) d x

Apply u-substitution

: \int \frac{sin ( u )}{2 π} d u

= \int \frac{sin ( u )}{2 π} d u

Take the constant out:

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

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

Use the common integral:

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

= \frac{1}{2 π} (- cos (u))

Substitute back

u = π x^{2}

= \frac{1}{2 π} (- cos (π x^{2}))

Simplify

= - \frac{1}{2 π} cos (π x^{2})

Add a constant to the solution

= - \frac{1}{2 π} cos (π x^{2}) + C

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