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

The integral of (x+5)/(sqrt(x)) is 2/3 x^{3/2}+10sqrt(x)+C

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

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

\frac{2}{3} x^{\frac{3}{2}} + 10 x + C

Solution steps

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

Expand

\frac{x + 5}{x} : x + \frac{5}{x}

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

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

\int \frac{5}{x} d x = 10 x

= \frac{2}{3} x^{\frac{3}{2}} + 10 x

Add a constant to the solution

= \frac{2}{3} x^{\frac{3}{2}} + 10 x + C

\int \frac{26 x ^{3} - 52 x ^{2} + 8}{x ^{2} - 2 x} d x

\int cot (16 x) d x

x \to 0 - lim (\frac{5}{x ^{3}})

x^{18}

\frac{d}{d x} (tan (2 x) cot (2 x))

What is the integral of (x+5)/(sqrt(x)) ?
The integral of (x+5)/(sqrt(x)) is 2/3 x^{3/2}+10sqrt(x)+C