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受欢迎的微积分 >

integral of (x^5)/(e^{2-3x^6)}

解答

\int \frac{x ^{5}}{e ^{2 - 3 x^{6}}} d x

解答

\frac{1}{18} e^{3 x^{6} - 2} + C

求解步骤

\int \frac{x ^{5}}{e ^{2 - 3 x^{6}}} d x

\frac{x ^{5}}{e ^{2 - 3 x^{6}}} = \frac{1}{e ^{2}} \frac{x ^{5}}{e ^{- 3 x^{6}}}

= \int \frac{1}{e ^{2}} \cdot \frac{x ^{5}}{e ^{- 3 x^{6}}} d x

提出常数:

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

= \frac{1}{e ^{2}} \cdot \int \frac{x ^{5}}{e ^{- 3 x^{6}}} d x

化简

\frac{x ^{5}}{e ^{- 3 x^{6}}} : e^{3 x^{6}} x^{5}

= \frac{1}{e ^{2}} \cdot \int e^{3 x^{6}} x^{5} d x

使用换元积分法

= \frac{1}{e ^{2}} \cdot \int \frac{e ^{3 u}}{6} d u

提出常数:

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

= \frac{1}{e ^{2}} \cdot \frac{1}{6} \cdot \int e^{3 u} d u

使用换元积分法

= \frac{1}{e ^{2}} \cdot \frac{1}{6} \cdot \int e^{v} \frac{1}{3} d v

提出常数:

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

= \frac{1}{e ^{2}} \cdot \frac{1}{6} \cdot \frac{1}{3} \cdot \int e^{v} d v

使用积分常见定则:

\int e^{v} d v = e^{v}

= \frac{1}{e ^{2}} \cdot \frac{1}{6} \cdot \frac{1}{3} e^{v}

代回

= \frac{1}{e ^{2}} \cdot \frac{1}{6} \cdot \frac{1}{3} e^{3 x^{6}}

化简

\frac{1}{e ^{2}} \cdot \frac{1}{6} \cdot \frac{1}{3} e^{3 x^{6}} : \frac{1}{18} e^{3 x^{6} - 2}

= \frac{1}{18} e^{3 x^{6} - 2}

解答补常数

= \frac{1}{18} e^{3 x^{6} - 2} + C

作图

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