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受欢迎的微积分 >

integral of r^2(r^3-4)^7

解答

\int r^{2} (r^{3} - 4)^{7} d r

解答

\frac{1}{24} (r^{3} - 4)^{8} + C

求解步骤

\int r^{2} (r^{3} - 4)^{7} d r

使用换元积分法

= \int \frac{u ^{7}}{3} d u

提出常数:

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

= \frac{1}{3} \cdot \int u^{7} d u

使用幂法则

= \frac{1}{3} \cdot \frac{u ^{8}}{8}

u = r^{3} - 4

代回

= \frac{1}{3} \cdot \frac{( r ^{3} - 4 ) ^{8}}{8}

化简

\frac{1}{3} \cdot \frac{( r ^{3} - 4 ) ^{8}}{8} : \frac{1}{24} (r^{3} - 4)^{8}

= \frac{1}{24} (r^{3} - 4)^{8}

解答补常数

= \frac{1}{24} (r^{3} - 4)^{8} + C

作图

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