What is the integral of (sqrt(5x)+sqrt(5/x)) ?

The integral of (sqrt(5x)+sqrt(5/x)) is sqrt(5) 2/3 x^{3/2}+sqrt(5/x)x+sqrt(5)sqrt(x)+C

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

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

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

Solution steps

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

Apply radical rule:

assuming

a \geq 0, b \geq 0

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

Apply the Sum Rule:

\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

Add a constant to the solution

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

What is the integral of (sqrt(5x)+sqrt(5/x)) ?
The integral of (sqrt(5x)+sqrt(5/x)) is sqrt(5) 2/3 x^{3/2}+sqrt(5/x)x+sqrt(5)sqrt(x)+C