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

The integral of 1/x (sqrt(x)+ln(x)) is 2sqrt(x)+(ln^2(x))/2+C

Popular >

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

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

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

Solution steps

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

Simplify

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

= \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

\int (x^{2} - x^{4}) d x

\int \frac{( x + 6 ) ^{2}}{x ^{3}} d x

\int \frac{2 x + 2}{x ^{2} + 2 x + 10} d x

\int \frac{8 x ^{3} + 16 x}{( x ^{2} + 4 ) ^{2}} d x

\int \frac{3 x ^{2} + 2 x + 3}{x ^{2} + 1} d x

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