What is the integral from 0 to pi of x|cos(x)| ?

The integral from 0 to pi of x|cos(x)| is pi

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integral from 0 to pi of x|cos(x)|

\int_{0}^{π} x ∣ cos (x) ∣ d x

π

Decimal

3.14159 \dots

Solution steps

\int_{0}^{π} x ∣ cos (x) ∣ d x

Eliminate Absolutes

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

\int_{0}^{\frac{π}{2}} x cos (x) d x = \frac{π}{2} - 1

\int_{\frac{π}{2}}^{π} x (- cos (x)) d x = 1 + \frac{π}{2}

= \frac{π}{2} - 1 + 1 + \frac{π}{2}

Simplify

\frac{π}{2} - 1 + 1 + \frac{π}{2} : π

= π

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

\int_{1}^{0} x - x^{2} d x

\int_{- 3}^{2} 2 x d x

\int_{0}^{8} 1 + x^{- \frac{2}{3}} d x

\int_{0}^{22 π} sin (x) d x

What is the integral from 0 to pi of x|cos(x)| ?
The integral from 0 to pi of x|cos(x)| is pi