What is the integral of 1/(2sqrt(x)\sqrt{x+1)} ?

The integral of 1/(2sqrt(x)\sqrt{x+1)} is ln|sqrt(x+1)+sqrt(x)|+C

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integral of 1/(2sqrt(x)\sqrt{x+1)}

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

ln x + 1 + x + C

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Apply u-substitution

: \int \frac{2}{u ^{2} + 1} d u

= \frac{1}{2} \cdot \int \frac{2}{u ^{2} + 1} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot 2 \cdot \int \frac{1}{u ^{2} + 1} d u

Use the common integral:

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

= \frac{1}{2} \cdot 2 ln u^{2} + 1 + u

Substitute back

u = x

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

Simplify

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

= ln x + 1 + x

Add a constant to the solution

= ln x + 1 + x + C

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