What is the integral of 1/(12sqrt(x)+12x) ?

The integral of 1/(12sqrt(x)+12x) is 1/6 ln|sqrt(x)+1|+C

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integral of 1/(12sqrt(x)+12x)

\int \frac{1}{12 x + 12 x} d x

\frac{1}{6} ln x + 1 + C

Solution steps

\int \frac{1}{12 x + 12 x} d x

Apply u-substitution

: \int \frac{1}{6 ( u + 1 )} d u

= \int \frac{1}{6 ( u + 1 )} d u

Take the constant out:

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

= \frac{1}{6} \cdot \int \frac{1}{u + 1} d u

Apply u-substitution

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

= \frac{1}{6} \cdot \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= \frac{1}{6} ln ∣ v ∣

Substitute back

= \frac{1}{6} ln x + 1

Add a constant to the solution

= \frac{1}{6} ln x + 1 + C

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What is the integral of 1/(12sqrt(x)+12x) ?
The integral of 1/(12sqrt(x)+12x) is 1/6 ln|sqrt(x)+1|+C