What is the integral of (2x+5)sqrt(x+3) ?

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

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integral of (2x+5)sqrt(x+3)

\int (2 x + 5) x + 3 d x

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

Solution steps

\int (2 x + 5) x + 3 d x

Expand

(2 x + 5) x + 3 : 2 x x + 3 + 5 x + 3

= \int 2 x x + 3 + 5 x + 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 2 x x + 3 d x + \int 5 x + 3 d x

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

\int 5 x + 3 d x = \frac{10}{3} (x + 3)^{\frac{3}{2}}

= 2 (\frac{2}{5} (x + 3)^{\frac{5}{2}} - 2 (x + 3)^{\frac{3}{2}}) + \frac{10}{3} (x + 3)^{\frac{3}{2}}

Simplify

2 (\frac{2}{5} (x + 3)^{\frac{5}{2}} - 2 (x + 3)^{\frac{3}{2}}) + \frac{10}{3} (x + 3)^{\frac{3}{2}} : \frac{4}{5} (x + 3)^{\frac{5}{2}} - \frac{2}{3} (x + 3)^{\frac{3}{2}}

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

Add a constant to the solution

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

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