What is the integral from 0 to pi/2 of 24x^2sin(x) ?

The integral from 0 to pi/2 of 24x^2sin(x) is 24(pi-2)

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integral from 0 to pi/2 of 24x^2sin(x)

\int_{0}^{\frac{π}{2}} 24 x^{2} sin (x) d x

24 (π - 2)

Decimal Notation

27.39822 \dots

Solution steps

\int_{0}^{\frac{π}{2}} 24 x^{2} sin (x) d x

Take the constant out:

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

= 24 \cdot \int_{0}^{\frac{π}{2}} x^{2} sin (x) d x

Apply Integration By Parts

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

= 24 [- x^{2} cos (x) - \int - 2 x cos (x) d x]_{0}^{\frac{π}{2}}

\int - 2 x cos (x) d x = - 2 (x sin (x) + cos (x))

= 24 [- x^{2} cos (x) - (- 2 (x sin (x) + cos (x)))]_{0}^{\frac{π}{2}}

Simplify

= 24 [- x^{2} cos (x) + 2 (x sin (x) + cos (x))]_{0}^{\frac{π}{2}}

Compute the boundaries:

π - 2

= 24 (π - 2)

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What is the integral from 0 to pi/2 of 24x^2sin(x) ?
The integral from 0 to pi/2 of 24x^2sin(x) is 24(pi-2)