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

The integral from 0 to 1 of xsin(n)pix is pi/3 sin(n)

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integral from 0 to 1 of xsin(n)pix

\int_{0}^{1} x sin (n) π x d x

\frac{π}{3} sin (n)

Solution steps

\int_{0}^{1} x sin (n) π x d x

Take the constant out:

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

= sin (n) π \cdot \int_{0}^{1} xx d x

Simplify

xx : x^{2}

= sin (n) π \cdot \int_{0}^{1} x^{2} d x

Apply the Power Rule

= sin (n) π [\frac{x ^{3}}{3}]_{0}^{1}

Compute the boundaries:

\frac{1}{3}

= sin (n) π \frac{1}{3}

Simplify

= \frac{π}{3} sin (n)

What is the integral from 0 to 1 of xsin(n)pix ?
The integral from 0 to 1 of xsin(n)pix is pi/3 sin(n)