What is the integral of xsqrt(2+x) ?

The integral of xsqrt(2+x) is 2/5 (2+x)^{5/2}-4/3 (2+x)^{3/2}+C

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integral of xsqrt(2+x)

\int x 2 + x d x

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

Solution steps

\int x 2 + x d x

Apply u-substitution

= \int (u - 2) u d u

Expand

(u - 2) u : u^{\frac{3}{2}} - 2 u

= \int u^{\frac{3}{2}} - 2 u d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int u^{\frac{3}{2}} d u - \int 2 u d u

\int u^{\frac{3}{2}} d u = \frac{2}{5} u^{\frac{5}{2}}

\int 2 u d u = \frac{4}{3} u^{\frac{3}{2}}

= \frac{2}{5} u^{\frac{5}{2}} - \frac{4}{3} u^{\frac{3}{2}}

Substitute back

u = 2 + x

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

Add a constant to the solution

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

What is the integral of xsqrt(2+x) ?
The integral of xsqrt(2+x) is 2/5 (2+x)^{5/2}-4/3 (2+x)^{3/2}+C