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受欢迎的微积分 >

integral of 3x^2(e^{x^3}-1)

解答

\int 3 x^{2} (e^{x^{3}} - 1) d x

解答

e^{x^{3}} - x^{3} + C

求解步骤

\int 3 x^{2} (e^{x^{3}} - 1) d x

提出常数:

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

= 3 \cdot \int x^{2} (e^{x^{3}} - 1) d x

使用换元积分法

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

提出常数:

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

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

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

= 3 \cdot \frac{1}{3} (\int e^{u} d u - \int 1 d u)

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

\int 1 d u = u

= 3 \cdot \frac{1}{3} (e^{u} - u)

u = x^{3}

代回

= 3 \cdot \frac{1}{3} (e^{x^{3}} - x^{3})

化简

3 \cdot \frac{1}{3} (e^{x^{3}} - x^{3}) : e^{x^{3}} - x^{3}

= e^{x^{3}} - x^{3}

解答补常数

= e^{x^{3}} - x^{3} + C

作图

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