What is the integral from 0 to 1 of xsin(pix^2) ?

The integral from 0 to 1 of xsin(pix^2) is 1/pi

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integral from 0 to 1 of xsin(pix^2)

\int_{0}^{1} x sin (π x^{2}) d x

\frac{1}{π}

Decimal

0.31830 \dots

Solution steps

\int_{0}^{1} x sin (π x^{2}) d x

Apply u-substitution

: \int_{0}^{π} \frac{sin ( u )}{2 π} d u

= \int_{0}^{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_{0}^{π} sin (u) d u

Use the common integral:

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

= \frac{1}{2 π} [- cos (u)]_{0}^{π}

Compute the boundaries:

2

= \frac{1}{2 π} \cdot 2

Simplify

= \frac{1}{π}

\int_{0}^{2} k x^{2} d x

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\int_{0}^{5} (5 - ∣ x - 5 ∣) d x

\int_{1}^{6} \frac{5}{x} d x

\int_{0}^{1} k x d x

What is the integral from 0 to 1 of xsin(pix^2) ?
The integral from 0 to 1 of xsin(pix^2) is 1/pi