What is the integral from 0 to pi/2 of x^3sin(x) ?

The integral from 0 to pi/2 of x^3sin(x) is 3((pi^2)/4-2)

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integral from 0 to pi/2 of x^3sin(x)

\int_{0}^{\frac{π}{2}} x^{3} sin (x) d x

3 (\frac{π ^{2}}{4} - 2)

Decimal

1.40220 \dots

Solution steps

\int_{0}^{\frac{π}{2}} x^{3} sin (x) d x

Apply Integration By Parts

: [- x^{3} cos (x) - \int - 3 x^{2} cos (x) d x]_{0}^{\frac{π}{2}}

= [- x^{3} cos (x) - \int - 3 x^{2} cos (x) d x]_{0}^{\frac{π}{2}}

\int - 3 x^{2} cos (x) d x = - 3 (x^{2} sin (x) - 2 (- x cos (x) + sin (x)))

= [- x^{3} cos (x) - (- 3 (x^{2} sin (x) - 2 (- x cos (x) + sin (x))))]_{0}^{\frac{π}{2}}

Simplify

= [- x^{3} cos (x) + 3 (x^{2} sin (x) - 2 (- x cos (x) + sin (x)))]_{0}^{\frac{π}{2}}

Compute the boundaries:

3 (\frac{π ^{2}}{4} - 2)

= 3 (\frac{π ^{2}}{4} - 2)

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What is the integral from 0 to pi/2 of x^3sin(x) ?
The integral from 0 to pi/2 of x^3sin(x) is 3((pi^2)/4-2)