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

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

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integral of x+sqrt(3x+1)

\int x + 3 x + 1 d x

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

Solution steps

\int x + 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 x d x + \int 3 x + 1 d x

\int x d x = \frac{x ^{2}}{2}

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

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

Add a constant to the solution

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

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