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

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

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integral of 1/(sqrt(x(x+2)))

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

ln x^{2} + 2 x + x + 1 + C

Solution steps

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

Complete the square

x (x + 2) : (x + 1)^{2} - 1

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

Apply u-substitution

= \int \frac{1}{u ^{2} - 1} d u

Apply Trigonometric Substitution

= \int sec (v) d v

Use the common integral:

\int sec (v) d v = ln ∣ tan (v) + sec (v) ∣

= ln ∣ tan (v) + sec (v) ∣

Substitute back

= ln ∣ tan (arcsec (x + 1)) + sec (arcsec (x + 1)) ∣

Simplify

ln ∣ tan (arcsec (x + 1)) + sec (arcsec (x + 1)) ∣ : ln x^{2} + 2 x + x + 1

= ln x^{2} + 2 x + x + 1

Add a constant to the solution

= ln x^{2} + 2 x + x + 1 + C

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