integral of sqrt(5x)+sqrt(5/x)

解答

\int 5 x + \frac{5}{x} d x

5 \frac{2}{3} x^{\frac{3}{2}} + \frac{5}{x} x + 5 x + C

求解步骤

\int 5 x + \frac{5}{x} d x

使用根式运算法则:

n ab = n a n b,

假定

a \geq 0, b \geq 0

= \int 5 x + \frac{5}{x} d x

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 5 x d x + \int \frac{5}{x} d x

\int 5 x d x = 5 \frac{2}{3} x^{\frac{3}{2}}

\int \frac{5}{x} d x = \frac{5}{x} x + 5 x

= 5 \frac{2}{3} x^{\frac{3}{2}} + \frac{5}{x} x + 5 x

解答补常数

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

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