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受欢迎的微积分 >

integral of sqrt(1+1/4 x)

解答

\int 1 + \frac{1}{4} x d x

解答

\frac{1}{3} (4 + x)^{\frac{3}{2}} + C

求解步骤

\int 1 + \frac{1}{4} x d x

化简

1 + \frac{1}{4} x : \frac{1}{2} 4 + x

= \int \frac{1}{2} 4 + x d x

提出常数:

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

= \frac{1}{2} \cdot \int 4 + x d x

使用换元积分法

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

使用幂法则

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

u = 4 + x

代回

= \frac{1}{2} \cdot \frac{2}{3} (4 + x)^{\frac{3}{2}}

化简

\frac{1}{2} \cdot \frac{2}{3} (4 + x)^{\frac{3}{2}} : \frac{1}{3} (4 + x)^{\frac{3}{2}}

= \frac{1}{3} (4 + x)^{\frac{3}{2}}

解答补常数

= \frac{1}{3} (4 + x)^{\frac{3}{2}} + C

作图

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