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受欢迎的微积分 >

integral of 5x^2(2x^3+6)^8

解答

\int 5 x^{2} (2 x^{3} + 6)^{8} d x

解答

\frac{5}{54} (2 x^{3} + 6)^{9} + C

求解步骤

\int 5 x^{2} (2 x^{3} + 6)^{8} d x

提出常数:

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

= 5 \cdot \int x^{2} (2 x^{3} + 6)^{8} d x

使用换元积分法

= 5 \cdot \int \frac{u ^{8}}{6} d u

提出常数:

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

= 5 \cdot \frac{1}{6} \cdot \int u^{8} d u

使用幂法则

= 5 \cdot \frac{1}{6} \cdot \frac{u ^{9}}{9}

u = 2 x^{3} + 6

代回

= 5 \cdot \frac{1}{6} \cdot \frac{( 2 x ^{3} + 6 ) ^{9}}{9}

化简

5 \cdot \frac{1}{6} \cdot \frac{( 2 x ^{3} + 6 ) ^{9}}{9} : \frac{5}{54} (2 x^{3} + 6)^{9}

= \frac{5}{54} (2 x^{3} + 6)^{9}

解答补常数

= \frac{5}{54} (2 x^{3} + 6)^{9} + C

作图

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