What is the integral of (x+6)/(sqrt(x)) ?

The integral of (x+6)/(sqrt(x)) is 2/3 x^{3/2}+12sqrt(x)+C

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integral of (x+6)/(sqrt(x))

\int \frac{x + 6}{x} d x

\frac{2}{3} x^{\frac{3}{2}} + 12 x + C

Solution steps

\int \frac{x + 6}{x} d x

Expand

\frac{x + 6}{x} : x + \frac{6}{x}

= \int x + \frac{6}{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{6}{x} d x

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

\int \frac{6}{x} d x = 12 x

= \frac{2}{3} x^{\frac{3}{2}} + 12 x

Add a constant to the solution

= \frac{2}{3} x^{\frac{3}{2}} + 12 x + C

y = 4 x + 9, (4, 5)

- 4 - 7 x^{2}

y^{^{''}} - 8 y^{^{'}} + 20 = 0

\frac{1}{2} sin (2 x)

\int_{- \infty}^{12} x e^{\frac{x}{3}} d x

What is the integral of (x+6)/(sqrt(x)) ?
The integral of (x+6)/(sqrt(x)) is 2/3 x^{3/2}+12sqrt(x)+C