What is the integral of (5x-x)/(sqrt(3x^2-9)) ?

The integral of (5x-x)/(sqrt(3x^2-9)) is 4/3 sqrt(3x^2-9)+C

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integral of (5x-x)/(sqrt(3x^2-9))

\int \frac{5 x - x}{3 x ^{2} - 9} d x

\frac{4}{3} 3 x^{2} - 9 + C

Solution steps

\int \frac{5 x - x}{3 x ^{2} - 9} d x

Expand

\frac{5 x - x}{3 x ^{2} - 9} : \frac{5 x}{3 x ^{2} - 9} - \frac{x}{3 x ^{2} - 9}

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{5 x}{3 x ^{2} - 9} d x - \int \frac{x}{3 x ^{2} - 9} d x

\int \frac{5 x}{3 x ^{2} - 9} d x = \frac{5}{3} 3 x^{2} - 9

\int \frac{x}{3 x ^{2} - 9} d x = \frac{1}{3} 3 x^{2} - 9

= \frac{5}{3} 3 x^{2} - 9 - \frac{1}{3} 3 x^{2} - 9

Simplify

\frac{5}{3} 3 x^{2} - 9 - \frac{1}{3} 3 x^{2} - 9 : \frac{4}{3} 3 x^{2} - 9

= \frac{4}{3} 3 x^{2} - 9

Add a constant to the solution

= \frac{4}{3} 3 x^{2} - 9 + C

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What is the integral of (5x-x)/(sqrt(3x^2-9)) ?
The integral of (5x-x)/(sqrt(3x^2-9)) is 4/3 sqrt(3x^2-9)+C