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受欢迎的微积分 >

integral of 3x^3e^{4x}

解答

\int 3 x^{3} e^{4 x} d x

解答

\frac{3}{256} (64 e^{4 x} x^{3} - 3 (16 e^{4 x} x^{2} - 2 (4 e^{4 x} x - e^{4 x}))) + C

求解步骤

\int 3 x^{3} e^{4 x} d x

提出常数:

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

= 3 \cdot \int x^{3} e^{4 x} d x

使用换元积分法

= 3 \cdot \int \frac{e ^{u} u ^{3}}{256} d u

提出常数:

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

= 3 \cdot \frac{1}{256} \cdot \int e^{u} u^{3} d u

使用分布积分法

= 3 \cdot \frac{1}{256} (u^{3} e^{u} - \int 3 u^{2} e^{u} d u)

\int 3 u^{2} e^{u} d u = 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u}))

= 3 \cdot \frac{1}{256} (u^{3} e^{u} - 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u})))

u = 4 x

代回

= 3 \cdot \frac{1}{256} ((4 x)^{3} e^{4 x} - 3 ((4 x)^{2} e^{4 x} - 2 (e^{4 x} \cdot 4 x - e^{4 x})))

化简

3 \cdot \frac{1}{256} ((4 x)^{3} e^{4 x} - 3 ((4 x)^{2} e^{4 x} - 2 (e^{4 x} \cdot 4 x - e^{4 x}))) : \frac{3}{256} (64 e^{4 x} x^{3} - 3 (16 e^{4 x} x^{2} - 2 (4 e^{4 x} x - e^{4 x})))

= \frac{3}{256} (64 e^{4 x} x^{3} - 3 (16 e^{4 x} x^{2} - 2 (4 e^{4 x} x - e^{4 x})))

解答补常数

= \frac{3}{256} (64 e^{4 x} x^{3} - 3 (16 e^{4 x} x^{2} - 2 (4 e^{4 x} x - e^{4 x}))) + C

作图

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