What is the integral from 0 to pi of-x^2+pix-sin(x) ?

The integral from 0 to pi of-x^2+pix-sin(x) is (pi^3)/6-2

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integral from 0 to pi of-x^2+pix-sin(x)

\int_{0}^{π} - x^{2} + π x - sin (x) d x

\frac{π ^{3}}{6} - 2

Decimal

3.16771 \dots

Solution steps

\int_{0}^{π} - x^{2} + π x - sin (x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int_{0}^{π} x^{2} d x = \frac{π ^{3}}{3}

\int_{0}^{π} π x d x = \frac{π ^{3}}{2}

\int_{0}^{π} sin (x) d x = 2

= - \frac{π ^{3}}{3} + \frac{π ^{3}}{2} - 2

Combine the fractions

- \frac{π ^{3}}{3} + \frac{π ^{3}}{2} : \frac{π ^{3}}{6}

= \frac{π ^{3}}{6} - 2

What is the integral from 0 to pi of-x^2+pix-sin(x) ?
The integral from 0 to pi of-x^2+pix-sin(x) is (pi^3)/6-2