What is the integral of (sqrt(x)+4x)/x ?

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

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integral of (sqrt(x)+4x)/x

\int \frac{x + 4 x}{x} d x

2 x + 4 x + C

Solution steps

\int \frac{x + 4 x}{x} d x

Simplify

\frac{x + 4 x}{x} : \frac{1 + 4 x}{x}

= \int \frac{1 + 4 x}{x} d x

Expand

\frac{1 + 4 x}{x} : \frac{1}{x} + 4

= \int \frac{1}{x} + 4 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 4 d x

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

\int 4 d x = 4 x

= 2 x + 4 x

Add a constant to the solution

= 2 x + 4 x + C

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