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

解答

\int_{0}^{π} (x - sin^{2} (x)) d x

\frac{π ^{2} - π}{2}

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3.36400 \dots

求解步骤

\int_{0}^{π} x - sin^{2} (x) d x

使用积分加法定则:

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

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

\int_{0}^{π} x d x = \frac{π ^{2}}{2}

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

= \frac{π ^{2}}{2} - \frac{π}{2}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{π ^{2} - π}{2}

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