What is the integral of (1+sqrt(x)+x)/(sqrt(x)) ?

The integral of (1+sqrt(x)+x)/(sqrt(x)) is 2sqrt(x)+x+2/3 x^{3/2}+C

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integral of (1+sqrt(x)+x)/(sqrt(x))

\int \frac{1 + x + x}{x} d x

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

Solution steps

\int \frac{1 + x + x}{x} d x

Expand

\frac{1 + x + x}{x} : \frac{1}{x} + 1 + x

= \int \frac{1}{x} + 1 + 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 \frac{1}{x} d x + \int 1 d x + \int x d x

\int \frac{1}{x} d x = 2 x

\int 1 d x = x

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

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

Add a constant to the solution

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

\frac{d y}{d x} = \frac{13 x}{y}

x ∣ x ∣

\int \frac{3 x}{1} d x

n = 1 \sum \infty 2 n + 1

x \to \frac{1}{2} lim (x tan (π) x)

What is the integral of (1+sqrt(x)+x)/(sqrt(x)) ?
The integral of (1+sqrt(x)+x)/(sqrt(x)) is 2sqrt(x)+x+2/3 x^{3/2}+C