integral from 0 to 1 of x^4e^{8x}

解答

\int_{0}^{1} x^{4} e^{8 x} d x

\frac{331 e ^{8} - 3}{4096}

十进制

240.89211 \dots

求解步骤

\int_{0}^{1} x^{4} e^{8 x} d x

使用分布积分法

= [\frac{1}{8} e^{8 x} x^{4} - \int \frac{1}{2} e^{8 x} x^{3} d x]_{0}^{1}

\int \frac{1}{2} e^{8 x} x^{3} d x = \frac{1}{8192} (512 e^{8 x} x^{3} - 3 (64 e^{8 x} x^{2} - 2 (8 e^{8 x} x - e^{8 x})))

= [\frac{1}{8} e^{8 x} x^{4} - \frac{1}{8192} (512 e^{8 x} x^{3} - 3 (64 e^{8 x} x^{2} - 2 (8 e^{8 x} x - e^{8 x})))]_{0}^{1}

化简

[\frac{1}{8} e^{8 x} x^{4} - \frac{1}{8192} (512 e^{8 x} x^{3} - 3 (64 e^{8 x} x^{2} - 2 (8 e^{8 x} x - e^{8 x})))]_{0}^{1} : [\frac{1}{4096} e^{8 x} (512 x^{4} - 256 x^{3} + 96 x^{2} - 24 x + 3)]_{0}^{1}

= [\frac{1}{4096} e^{8 x} (512 x^{4} - 256 x^{3} + 96 x^{2} - 24 x + 3)]_{0}^{1}

带入上下限计算后得:

\frac{331 e ^{8}}{4096} - \frac{3}{4096}

= \frac{331 e ^{8}}{4096} - \frac{3}{4096}

化简

= \frac{331 e ^{8} - 3}{4096}

d er i v a t i v e f (x) = (x - 2) (2 x + 3)

d er i v a t i v e f (t) = 3 t^{3} - \frac{2}{t ^{2}}

x \to 0 + lim (\frac{1}{2 x ^{2}})

d er i v a t i v e 3 + 5 x

\frac{d}{d x} (x + y z)