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

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

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

\int \frac{( x + ln ( x ) )}{x} d x

2 x + \frac{ln ^{2} ( x )}{2} + C

Solution steps

\int \frac{x + ln ( x )}{x} d x

Expand

\frac{x + ln ( x )}{x} : \frac{1}{x} + \frac{ln ( x )}{x}

= \int \frac{1}{x} + \frac{ln ( x )}{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 \frac{ln ( x )}{x} d x

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

\int \frac{ln ( x )}{x} d x = \frac{ln ^{2} ( x )}{2}

= 2 x + \frac{ln ^{2} ( x )}{2}

Add a constant to the solution

= 2 x + \frac{ln ^{2} ( x )}{2} + C

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