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

The integral from 0 to pi/2 of 2pixcos(x) is pi^2-2pi

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integral from 0 to pi/2 of 2pixcos(x)

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

π^{2} - 2 π

Solution steps

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

Take the constant out:

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

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

Apply Integration By Parts

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

= 2 π [x sin (x) - \int sin (x) d x]_{0}^{\frac{π}{2}}

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

= 2 π [x sin (x) - (- cos (x))]_{0}^{\frac{π}{2}}

Simplify

= 2 π [x sin (x) + cos (x)]_{0}^{\frac{π}{2}}

Compute the boundaries:

\frac{π}{2} - 1

= 2 π (\frac{π}{2} - 1)

Simplify

= π^{2} - 2 π

\int_{3}^{6} x^{2} - 3 x d x

\int_{0}^{3} 6 x - x^{2} d x

\int_{- \frac{π}{2}}^{\frac{π}{2}} cos (x) - 1 d x

\int_{0}^{2 π} \frac{1}{2}

\int_{- \frac{π}{2}}^{\frac{π}{2}} cos (x) - 4 d x

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