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

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

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

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

= 2 x 1 + x - (\frac{1 + x}{x} x + ln x + 1 + x)

Simplify

= 2 x 1 + x - \frac{1 + x}{x} x - ln x + 1 + x

Add a constant to the solution

= 2 x 1 + x - \frac{1 + x}{x} x - ln x + 1 + x + C

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