integral from a to b of (x^2)/(b-a)

解答

\int_{a}^{b} \frac{x ^{2}}{b - a} d x

\frac{b ^{2} + ba + a ^{2}}{3}

求解步骤

\int_{a}^{b} \frac{x ^{2}}{b - a} d x

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{b - a} \cdot \int_{a}^{b} x^{2} d x

使用幂法则

= \frac{1}{b - a} [\frac{x ^{3}}{3}]_{a}^{b}

化简

\frac{1}{b - a} [\frac{x ^{3}}{3}]_{a}^{b} : \frac{[ \frac{x ^{3}}{3} ] _{a}^{b}}{b - a}

= \frac{[ \frac{x ^{3}}{3} ] _{a}^{b}}{b - a}

带入上下限计算后得:

\frac{b ^{3}}{3} - \frac{a ^{3}}{3}

= \frac{\frac{b ^{3}}{3} - \frac{a ^{3}}{3}}{b - a}

化简

= \frac{b ^{2} + ba + a ^{2}}{3}

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