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

The integral from 0 to pi/2 of xsin(x^2) is 1/2 (-cos((pi^2)/4)+1)

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integral from 0 to pi/2 of xsin(x^2)

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

\frac{1}{2} (- cos (\frac{π ^{2}}{4}) + 1)

Decimal

0.89060 \dots

Solution steps

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

Apply u-substitution

: \int_{0}^{\frac{π ^{2}}{4}} \frac{sin ( u )}{2} d u

= \int_{0}^{\frac{π ^{2}}{4}} \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_{0}^{\frac{π ^{2}}{4}} sin (u) d u

Use the common integral:

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

= \frac{1}{2} [- cos (u)]_{0}^{\frac{π ^{2}}{4}}

Compute the boundaries:

- cos (\frac{π ^{2}}{4}) + 1

= \frac{1}{2} (- cos (\frac{π ^{2}}{4}) + 1)

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What is the integral from 0 to pi/2 of xsin(x^2) ?
The integral from 0 to pi/2 of xsin(x^2) is 1/2 (-cos((pi^2)/4)+1)