What is the integral from 0 to pi of (pi-x)cos(n)x ?

The integral from 0 to pi of (pi-x)cos(n)x is (pi^3)/6 cos(n)

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integral from 0 to pi of (pi-x)cos(n)x

\int_{0}^{π} (π - x) cos (n) x d x

\frac{π ^{3}}{6} cos (n)

Solution steps

\int_{0}^{π} (π - x) cos (n) x d x

Take the constant out:

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

= cos (n) \cdot \int_{0}^{π} (π - x) x d x

Expand

(π - x) x : π x - x^{2}

= cos (n) \cdot \int_{0}^{π} π x - x^{2} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= cos (n) (\int_{0}^{π} π x d x - \int_{0}^{π} x^{2} d x)

\int_{0}^{π} π x d x = \frac{π ^{3}}{2}

\int_{0}^{π} x^{2} d x = \frac{π ^{3}}{3}

= cos (n) (\frac{π ^{3}}{2} - \frac{π ^{3}}{3})

Join

\frac{π ^{3}}{2} - \frac{π ^{3}}{3} : \frac{π ^{3}}{6}

= \frac{π ^{3}}{6} cos (n)

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What is the integral from 0 to pi of (pi-x)cos(n)x ?
The integral from 0 to pi of (pi-x)cos(n)x is (pi^3)/6 cos(n)