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

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

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

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

\frac{1 + x}{x} x + ln x + 1 + x + C

Solution steps

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

Apply Integration By Parts

: \frac{1 + x}{x} x - \int - \frac{1}{2 x ^{\frac{1}{2}} x + 1} d x

= \frac{1 + x}{x} x - \int - \frac{1}{2 x ^{\frac{1}{2}} x + 1} d x

\int - \frac{1}{2 x ^{\frac{1}{2}} x + 1} d x = - ln x + 1 + x

= \frac{1 + x}{x} x - (- ln x + 1 + x)

Simplify

= \frac{1 + x}{x} x + ln x + 1 + x

Add a constant to the solution

= \frac{1 + x}{x} x + ln x + 1 + x + C

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