integral from x^4 to x^5 of (-6t-1)^4

解答

\int_{x^{4}}^{x^{5}} (- 6 t - 1)^{4} d t

\frac{x ^{4} ( 1296 x ^{21} - 216 x ^{16} - 1080 x ^{12} + 360 x ^{11} - 360 x ^{8} + 60 x ^{6} - 60 x ^{4} + 5 x - 5 )}{5}

求解步骤

\int_{x^{4}}^{x^{5}} (- 6 t - 1)^{4} d t

因式分解

- 6 t - 1 : - (6 t + 1)

= \int_{x^{4}}^{x^{5}} (- (6 t + 1))^{4} d t

化简

(- (6 t + 1))^{4} : (6 t + 1)^{4}

= \int_{x^{4}}^{x^{5}} (6 t + 1)^{4} d t

使用换元积分法

= \int_{6 x^{4} + 1}^{6 x^{5} + 1} \frac{u ^{4}}{6} d u

提出常数:

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

= \frac{1}{6} \cdot \int_{6 x^{4} + 1}^{6 x^{5} + 1} u^{4} d u

使用幂法则

= \frac{1}{6} [\frac{u ^{5}}{5}]_{6 x^{4} + 1}^{6 x^{5} + 1}

带入上下限计算后得:

\frac{( 6 x ^{5} + 1 ) ^{5}}{5} - \frac{( 6 x ^{4} + 1 ) ^{5}}{5}

= \frac{1}{6} (\frac{( 6 x ^{5} + 1 ) ^{5}}{5} - \frac{( 6 x ^{4} + 1 ) ^{5}}{5})

化简

= \frac{x ^{4} ( 1296 x ^{21} - 216 x ^{16} - 1080 x ^{12} + 360 x ^{11} - 360 x ^{8} + 60 x ^{6} - 60 x ^{4} + 5 x - 5 )}{5}

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