integral from a to b of ((ax)^3+e^{-bx})

解答

\int_{a}^{b} ((a x)^{3} + e^{- b x}) d x

\frac{a ^{3} ( b ^{4} - a ^{4} )}{4} - \frac{e ^{- b^{2}} - e ^{- ab}}{b}

求解步骤

\int_{a}^{b} (a x)^{3} + e^{- b x} d x

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

= \int_{a}^{b} a^{3} x^{3} + e^{- b 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_{a}^{b} a^{3} x^{3} d x + \int_{a}^{b} e^{- b x} d x

\int_{a}^{b} a^{3} x^{3} d x = \frac{a ^{3} ( b ^{4} - a ^{4} )}{4}

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

= \frac{a ^{3} ( b ^{4} - a ^{4} )}{4} - \frac{e ^{- b^{2}} - e ^{- ab}}{b}

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