What is the integral of 1/((x+2)sqrt(x^2+4x)) ?

The integral of 1/((x+2)sqrt(x^2+4x)) is 1/2 arcsec(1/2 x+1)+C

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integral of 1/((x+2)sqrt(x^2+4x))

\int \frac{1}{( x + 2 ) x ^{2} + 4 x} d x

\frac{1}{2} arcsec (\frac{1}{2} x + 1) + C

Solution steps

\int \frac{1}{( x + 2 ) x ^{2} + 4 x} d x

Apply u-substitution

: \int \frac{1}{u u ^{2} - 4} d u

= \int \frac{1}{u u ^{2} - 4} d u

Apply Trigonometric Substitution

: \int \frac{1}{2} d v

= \int \frac{1}{2} d v

Take the constant out:

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

= \frac{1}{2} \cdot \int 1 d v

Integral of a constant:

\int a d x = a x

= \frac{1}{2} \cdot 1 \cdot v

Substitute back

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

Simplify

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

= \frac{1}{2} arcsec (\frac{1}{2} x + 1)

Add a constant to the solution

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

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What is the integral of 1/((x+2)sqrt(x^2+4x)) ?
The integral of 1/((x+2)sqrt(x^2+4x)) is 1/2 arcsec(1/2 x+1)+C