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

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

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

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

2 (ln x - ln 2 x + 1) + C

Solution steps

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

Apply u-substitution

: \int \frac{2}{u ( 2 u + 1 )} d u

= \int \frac{2}{u ( 2 u + 1 )} d u

Take the constant out:

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

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

Take the partial fraction of

\frac{1}{u ( 2 u + 1 )} : \frac{1}{u} - \frac{2}{2 u + 1}

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int \frac{1}{u} d u = ln ∣ u ∣

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

= 2 (ln ∣ u ∣ - ln ∣ 2 u + 1 ∣)

Substitute back

u = x

= 2 (ln x - ln 2 x + 1)

Add a constant to the solution

= 2 (ln x - ln 2 x + 1) + C

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