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

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

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

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

π - 2

Decimal

1.14159 \dots

Solution steps

\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}}

= [- 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))

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

Simplify

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

Compute the boundaries:

π - 2

= π - 2

\int_{2}^{6} 4 x + 1 d x

\int_{1}^{2} \frac{4}{x ^{3}} d x

\int_{1}^{10} x \frac{2}{81} (x - 1) d x

\int_{3}^{\infty} e^{- s t} d t

\int_{0}^{9} (81 - x^{2}) d x

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