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

The integral of (x*sqrt(x+2)+2*sqrt(x+2)) is 2/5 (x+2)^{5/2}+C

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integral of (xsqrt(x+2)+2sqrt(x+2))

\int (x \cdot x + 2 + 2 \cdot x + 2) d x

\frac{2}{5} (x + 2)^{\frac{5}{2}} + C

Solution steps

\int x x + 2 + 2 x + 2 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 x + 2 d x + \int 2 x + 2 d x

\int x x + 2 d x = \frac{2}{5} (x + 2)^{\frac{5}{2}} - \frac{4}{3} (x + 2)^{\frac{3}{2}}

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

= \frac{2}{5} (x + 2)^{\frac{5}{2}} - \frac{4}{3} (x + 2)^{\frac{3}{2}} + \frac{4}{3} (x + 2)^{\frac{3}{2}}

Add similar elements:

- \frac{4}{3} (x + 2)^{\frac{3}{2}} + \frac{4}{3} (x + 2)^{\frac{3}{2}} = 0

= \frac{2}{5} (x + 2)^{\frac{5}{2}}

Add a constant to the solution

= \frac{2}{5} (x + 2)^{\frac{5}{2}} + C

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What is the integral of (x*sqrt(x+2)+2*sqrt(x+2)) ?
The integral of (x*sqrt(x+2)+2*sqrt(x+2)) is 2/5 (x+2)^{5/2}+C