What is the integral from 1 to 2 of xsin(pix^2) ?

The integral from 1 to 2 of xsin(pix^2) is -1/pi

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integral from 1 to 2 of xsin(pix^2)

\int_{1}^{2} x sin (π x^{2}) d x

- \frac{1}{π}

Decimal

- 0.31830 \dots

Solution steps

\int_{1}^{2} x sin (π x^{2}) d x

Apply u-substitution

: \int_{π}^{4 π} \frac{sin ( u )}{2 π} d u

= \int_{π}^{4 π} \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_{π}^{4 π} sin (u) d u

Use the common integral:

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

= \frac{1}{2 π} [- cos (u)]_{π}^{4 π}

Compute the boundaries:

- 2

= \frac{1}{2 π} (- 2)

Simplify

= - \frac{1}{π}

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What is the integral from 1 to 2 of xsin(pix^2) ?
The integral from 1 to 2 of xsin(pix^2) is -1/pi