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

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

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

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

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

Solution steps

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

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

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

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

ln (x^{6})

n = 0 \sum \infty \frac{4 ^{(2 n + 1)}}{( - 4 ) ^{n}}

\frac{d}{d x} (- \frac{1}{4 x ^{\frac{3}{2}}})

\int e^{5 x + 3} d x

\int \frac{2 ^{t a n (x)}}{cos ^{2} ( x )} d x

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