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受欢迎的微积分 >

integral of (2x)/(sqrt(8+x^2))

解答

\int \frac{2 x}{8 + x ^{2}} d x

解答

2 8 + x^{2} + C

求解步骤

\int \frac{2 x}{8 + x ^{2}} d x

提出常数:

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

= 2 \cdot \int \frac{x}{8 + x ^{2}} d x

使用换元积分法

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

使用幂法则

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

u = 8 + x^{2}

代回

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

化简

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

= 2 8 + x^{2}

解答补常数

= 2 8 + x^{2} + C

作图

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