What is the integral of (sqrt(3x+1)-x/(x^2+3)) ?

The integral of (sqrt(3x+1)-x/(x^2+3)) is 2/9 (3x+1)^{3/2}-1/2 ln|x^2+3|+C

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integral of (sqrt(3x+1)-x/(x^2+3))

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

\frac{2}{9} (3 x + 1)^{\frac{3}{2}} - \frac{1}{2} ln x^{2} + 3 + C

Solution steps

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

\int 3 x + 1 d x = \frac{2}{9} (3 x + 1)^{\frac{3}{2}}

\int \frac{x}{x ^{2} + 3} d x = \frac{1}{2} ln x^{2} + 3

= \frac{2}{9} (3 x + 1)^{\frac{3}{2}} - \frac{1}{2} ln x^{2} + 3

Add a constant to the solution

= \frac{2}{9} (3 x + 1)^{\frac{3}{2}} - \frac{1}{2} ln x^{2} + 3 + C

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What is the integral of (sqrt(3x+1)-x/(x^2+3)) ?
The integral of (sqrt(3x+1)-x/(x^2+3)) is 2/9 (3x+1)^{3/2}-1/2 ln|x^2+3|+C