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

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

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

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

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

Solution steps

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

Simplify

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

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

\frac{d}{d x} (2 x^{2} cos (x) cot (x))

\frac{d y}{d x}, x y - 8 = 0

\frac{d}{d x} (3 a + b x^{3})

f (x) = 5 x

\int_{1}^{4} x (1 + x) d x

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