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受欢迎的微积分 >

integral of (x^2+4)^72x

解答

\int (x^{2} + 4)^{7} 2 x d x

解答

\frac{1}{8} (x^{2} + 4)^{8} + C

求解步骤

\int (x^{2} + 4)^{7} \cdot 2 x d x

提出常数:

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

= 2 \cdot \int (x^{2} + 4)^{7} x d x

使用换元积分法

= 2 \cdot \int \frac{u ^{7}}{2} d u

提出常数:

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

= 2 \cdot \frac{1}{2} \cdot \int u^{7} d u

使用幂法则

= 2 \cdot \frac{1}{2} \cdot \frac{u ^{8}}{8}

u = x^{2} + 4

代回

= 2 \cdot \frac{1}{2} \cdot \frac{( x ^{2} + 4 ) ^{8}}{8}

化简

2 \cdot \frac{1}{2} \cdot \frac{( x ^{2} + 4 ) ^{8}}{8} : \frac{1}{8} (x^{2} + 4)^{8}

= \frac{1}{8} (x^{2} + 4)^{8}

解答补常数

= \frac{1}{8} (x^{2} + 4)^{8} + C

作图

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