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受欢迎的微积分 >

integral of sqrt(1+5x^2)(10x)

解答

\int 1 + 5 x^{2} (10 x) d x

解答

\frac{2}{3} (1 + 5 x^{2})^{\frac{3}{2}} + C

求解步骤

\int 1 + 5 x^{2} \cdot 10 x d x

提出常数:

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

= 10 \cdot \int 1 + 5 x^{2} x d x

使用换元积分法

= 10 \cdot \int \frac{u}{10} d u

提出常数:

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

= 10 \cdot \frac{1}{10} \cdot \int u d u

使用幂法则

= 10 \cdot \frac{1}{10} \cdot \frac{2}{3} u^{\frac{3}{2}}

u = 1 + 5 x^{2}

代回

= 10 \cdot \frac{1}{10} \cdot \frac{2}{3} (1 + 5 x^{2})^{\frac{3}{2}}

化简

10 \cdot \frac{1}{10} \cdot \frac{2}{3} (1 + 5 x^{2})^{\frac{3}{2}} : \frac{2}{3} (1 + 5 x^{2})^{\frac{3}{2}}

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

解答补常数

= \frac{2}{3} (1 + 5 x^{2})^{\frac{3}{2}} + C

作图

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