What is the integral from 0 to pi/2 of xcos(7x) ?

The integral from 0 to pi/2 of xcos(7x) is (-7pi-2)/(98)

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integral from 0 to pi/2 of xcos(7x)

\int_{0}^{\frac{π}{2}} x cos (7 x) d x

\frac{- 7 π - 2}{98}

Decimal

- 0.24480 \dots

Solution steps

\int_{0}^{\frac{π}{2}} x cos (7 x) d x

Apply u-substitution

: \int_{0}^{\frac{7 π}{2}} \frac{u cos ( u )}{49} d u

= \int_{0}^{\frac{7 π}{2}} \frac{u cos ( u )}{49} d u

Take the constant out:

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

= \frac{1}{49} \cdot \int_{0}^{\frac{7 π}{2}} u cos (u) d u

Apply Integration By Parts

: [u sin (u) - \int sin (u) d u]_{0}^{\frac{7 π}{2}}

= \frac{1}{49} [u sin (u) - \int sin (u) d u]_{0}^{\frac{7 π}{2}}

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

= \frac{1}{49} [u sin (u) - (- cos (u))]_{0}^{\frac{7 π}{2}}

Simplify

= \frac{1}{49} [u sin (u) + cos (u)]_{0}^{\frac{7 π}{2}}

Compute the boundaries:

- \frac{7 π}{2} - 1

= \frac{1}{49} (- \frac{7 π}{2} - 1)

Simplify

= \frac{- 7 π - 2}{98}

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What is the integral from 0 to pi/2 of xcos(7x) ?
The integral from 0 to pi/2 of xcos(7x) is (-7pi-2)/(98)