What is the integral of (sqrt(x))/(6sqrt(x^3)+2x) ?

The integral of (sqrt(x))/(6sqrt(x^3)+2x) is 1/3 ln|3sqrt(x)+1|+C

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integral of (sqrt(x))/(6sqrt(x^3)+2x)

\int \frac{x}{6 x ^{3} + 2 x} d x

\frac{1}{3} ln 3 x + 1 + C

Solution steps

\int \frac{x}{6 x ^{3} + 2 x} d x

Apply u-substitution

: \int \frac{1}{3 u + 1} d u

= \int \frac{1}{3 u + 1} d u

Apply u-substitution

: \int \frac{1}{3 v} d v

= \int \frac{1}{3 v} d v

Take the constant out:

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

= \frac{1}{3} \cdot \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= \frac{1}{3} ln ∣ v ∣

Substitute back

= \frac{1}{3} ln 3 x + 1

Add a constant to the solution

= \frac{1}{3} ln 3 x + 1 + C

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What is the integral of (sqrt(x))/(6sqrt(x^3)+2x) ?
The integral of (sqrt(x))/(6sqrt(x^3)+2x) is 1/3 ln|3sqrt(x)+1|+C