What is the integral of 2+sqrt(4-x)+x ?

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

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integral of 2+sqrt(4-x)+x

\int 2 + 4 - x + x d x

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

Solution steps

\int 2 + 4 - x + x 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 2 d x + \int 4 - x d x + \int x d x

\int 2 d x = 2 x

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

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

= 2 x - \frac{2}{3} (4 - x)^{\frac{3}{2}} + \frac{x ^{2}}{2}

Add a constant to the solution

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

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