What is the integral of ((3x+1)/(sqrt(x))) ?

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

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integral of ((3x+1)/(sqrt(x)))

\int (\frac{3 x + 1}{x}) d x

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

Solution steps

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

Expand

\frac{3 x + 1}{x} : 3 x + \frac{1}{x}

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

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

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

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

Add a constant to the solution

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

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