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

The integral from 0 to pi/4 of x^2cos(x) is (pi^2)/(16sqrt(2))-2(1/(sqrt(2))-pi/(4sqrt(2)))

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

\int_{0}^{\frac{π}{4}} x^{2} cos (x) d x

\frac{π ^{2}}{16 2} - 2 (\frac{1}{2} - \frac{π}{4 2})

Decimal Notation

0.13268 \dots

Solution steps

\int_{0}^{\frac{π}{4}} x^{2} cos (x) d x

Apply Integration By Parts

: [x^{2} sin (x) - \int 2 x sin (x) d x]_{0}^{\frac{π}{4}}

= [x^{2} sin (x) - \int 2 x sin (x) d x]_{0}^{\frac{π}{4}}

\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}{4}}

Compute the boundaries:

\frac{π ^{2}}{16 2} - 2 (\frac{1}{2} - \frac{π}{4 2})

= \frac{π ^{2}}{16 2} - 2 (\frac{1}{2} - \frac{π}{4 2})

\int_{z}^{2 - z^{2}} z d z

\int_{0}^{t} 5 d t

\int_{0}^{1} e^{- x} - e^{x} d x

\int_{0}^{1} 2 π x (x - x^{4}) d x

\int_{- 1}^{1} (y^{2} - 2 - e^{y}) d y

What is the integral from 0 to pi/4 of x^2cos(x) ?
The integral from 0 to pi/4 of x^2cos(x) is (pi^2)/(16sqrt(2))-2(1/(sqrt(2))-pi/(4sqrt(2)))