What is the integral of ((2x+5))/(sqrt(x)(x+5)) ?

The integral of ((2x+5))/(sqrt(x)(x+5)) is -2sqrt(5)arctan(sqrt(x/5))+4sqrt(5)sqrt(x/5)+C

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integral of ((2x+5))/(sqrt(x)(x+5))

\int \frac{( 2 x + 5 )}{x ( x + 5 )} d x

- 25 arctan (\frac{x}{5}) + 45 \frac{x}{5} + C

Solution steps

\int \frac{2 x + 5}{x ( x + 5 )} d x

Expand

\frac{2 x + 5}{x ( x + 5 )} : \frac{2 x}{x ( x + 5 )} + \frac{5}{x ( x + 5 )}

Apply the Sum Rule:

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

= \int \frac{2 x}{x ( x + 5 )} d x + \int \frac{5}{x ( x + 5 )} d x

\int \frac{2 x}{x ( x + 5 )} d x = 45 (- arctan (\frac{x}{5}) + \frac{x}{5})

\int \frac{5}{x ( x + 5 )} d x = 25 arctan (\frac{x}{5})

= 45 (- arctan (\frac{x}{5}) + \frac{x}{5}) + 25 arctan (\frac{x}{5})

Simplify

45 (- arctan (\frac{x}{5}) + \frac{x}{5}) + 25 arctan (\frac{x}{5}) : - 25 arctan (\frac{x}{5}) + 45 \frac{x}{5}

= - 25 arctan (\frac{x}{5}) + 45 \frac{x}{5}

Add a constant to the solution

= - 25 arctan (\frac{x}{5}) + 45 \frac{x}{5} + C

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What is the integral of ((2x+5))/(sqrt(x)(x+5)) ?
The integral of ((2x+5))/(sqrt(x)(x+5)) is -2sqrt(5)arctan(sqrt(x/5))+4sqrt(5)sqrt(x/5)+C