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

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

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

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

\frac{π ^{2}}{4} - 2

Decimal

0.46740 \dots

Solution steps

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

Apply Integration By Parts

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

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

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

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

Compute the boundaries:

\frac{π ^{2}}{4} - 2

= \frac{π ^{2}}{4} - 2

\int_{0}^{8} \frac{1}{2 ( x ^{\frac{1}{3}} )} d x

\int_{1}^{\infty} \frac{1}{x ^{p}} d x

\int_{- 1}^{1} \frac{sin ( x )}{1 + x ^{2}} d x

\int_{0}^{π} 10 sin (x) d x

\int_{0}^{x} t cos (t) d t

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