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受欢迎的微积分 >

integral of 3x^2(x^3+5)^7

解答

\int 3 x^{2} (x^{3} + 5)^{7} d x

解答

\frac{1}{8} (x^{3} + 5)^{8} + C

求解步骤

\int 3 x^{2} (x^{3} + 5)^{7} d x

提出常数:

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

= 3 \cdot \int x^{2} (x^{3} + 5)^{7} d x

使用换元积分法

= 3 \cdot \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

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

使用幂法则

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

u = x^{3} + 5

代回

= 3 \cdot \frac{1}{3} \cdot \frac{( x ^{3} + 5 ) ^{8}}{8}

化简

3 \cdot \frac{1}{3} \cdot \frac{( x ^{3} + 5 ) ^{8}}{8} : \frac{1}{8} (x^{3} + 5)^{8}

= \frac{1}{8} (x^{3} + 5)^{8}

解答补常数

= \frac{1}{8} (x^{3} + 5)^{8} + C

作图

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