What is the integral from 0 to pi/4 of xsin(x) ?

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

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integral from 0 to pi/4 of xsin(x)

\int_{0}^{\frac{π}{4}} x sin (x) d x

\frac{1}{2} - \frac{π}{4 2}

Decimal

0.15174 \dots

Solution steps

\int_{0}^{\frac{π}{4}} x sin (x) d x

Apply Integration By Parts

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

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

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

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

Simplify

= [- x cos (x) + sin (x)]_{0}^{\frac{π}{4}}

Compute the boundaries:

\frac{1}{2} - \frac{π}{4 2}

= \frac{1}{2} - \frac{π}{4 2}

\int_{0}^{2 π} x cos (x) d x

\int_{- 2}^{1} (3 + y^{2}) - (2 - y^{2}) d y

\int_{0}^{\frac{π}{6}} sec (x) tan (x) d x

\int_{- 2}^{1} ∣ x + 1 ∣ d x

\int_{2}^{\infty} \frac{4}{x ^{2}} d x

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