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

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

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

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

\frac{2}{3} x - \frac{2}{3} ln 3 x + 1 + 1 + C

Solution steps

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

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

= \int - \frac{3 x}{1 + 3 x} + 1 d x

Apply the Sum Rule:

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

= - \int \frac{3 x}{1 + 3 x} d x + \int 1 d x

\int \frac{3 x}{1 + 3 x} d x = \frac{1}{3} (3 x - 23 x + 2 ln 3 x + 1 - 3)

\int 1 d x = x

= - \frac{1}{3} (3 x - 23 x + 2 ln 3 x + 1 - 3) + x

Simplify

- \frac{1}{3} (3 x - 23 x + 2 ln 3 x + 1 - 3) + x : \frac{2}{3} x - \frac{2}{3} ln 3 x + 1 + 1

= \frac{2}{3} x - \frac{2}{3} ln 3 x + 1 + 1

Add a constant to the solution

= \frac{2}{3} x - \frac{2}{3} ln 3 x + 1 + 1 + C

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