integral of r^2((r^3)/(21)+8)^6

解答

\int r^{2} (\frac{r ^{3}}{21} + 8)^{6} d r

(\frac{r ^{3}}{21} + 8)^{7} + C

求解步骤

\int r^{2} (\frac{r ^{3}}{21} + 8)^{6} d r

使用换元积分法

= \int 7 u^{6} d u

提出常数:

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

= 7 \cdot \int u^{6} d u

使用幂法则

= 7 \cdot \frac{u ^{7}}{7}

u = \frac{r ^{3}}{21} + 8

代回

= 7 \cdot \frac{( \frac{r ^{3}}{21} + 8 ) ^{7}}{7}

化简

7 \cdot \frac{( \frac{r ^{3}}{21} + 8 ) ^{7}}{7} : (\frac{r ^{3}}{21} + 8)^{7}

= (\frac{r ^{3}}{21} + 8)^{7}

解答补常数

= (\frac{r ^{3}}{21} + 8)^{7} + C

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