What is the integral of 1/(sqrt(3x)(1+3x)) ?

The integral of 1/(sqrt(3x)(1+3x)) is 2/3 arctan(sqrt(3)sqrt(x))+C

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integral of 1/(sqrt(3x)(1+3x))

\int \frac{1}{3 x ( 1 + 3 x )} d x

\frac{2}{3} arctan (3 x) + C

Solution steps

\int \frac{1}{3 x ( 1 + 3 x )} d x

3 x = 3 x

= \int \frac{1}{3 x ( 1 + 3 x )} d x

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}{x ( 1 + 3 x )} d x

Apply u-substitution

= \frac{1}{3} \cdot \int \frac{2}{1 + 3 u ^{2}} d u

Take the constant out:

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

= \frac{1}{3} \cdot 2 \cdot \int \frac{1}{1 + 3 u ^{2}} d u

Apply Integral Substitution

= \frac{1}{3} \cdot 2 \cdot \int \frac{1}{3 ( 1 + v ^{2} )} d v

Take the constant out:

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

= \frac{1}{3} \cdot 2 \cdot \frac{1}{3} \cdot \int \frac{1}{1 + v ^{2}} d v

Use the common integral:

\int \frac{1}{1 + v ^{2}} d v = arctan (v)

= \frac{1}{3} \cdot 2 \cdot \frac{1}{3} arctan (v)

Substitute back

= \frac{1}{3} \cdot 2 \cdot \frac{1}{3} arctan (3 x)

Simplify

\frac{1}{3} \cdot 2 \cdot \frac{1}{3} arctan (3 x) : \frac{2}{3} arctan (3 x)

= \frac{2}{3} arctan (3 x)

Add a constant to the solution

= \frac{2}{3} arctan (3 x) + C

What is the integral of 1/(sqrt(3x)(1+3x)) ?
The integral of 1/(sqrt(3x)(1+3x)) is 2/3 arctan(sqrt(3)sqrt(x))+C