What is the integral of (x^2+x+4)sqrt(x+3) ?

The integral of (x^2+x+4)sqrt(x+3) is 2/7 (x+3)^{7/2}-2(x+3)^{5/2}+20/3 (x+3)^{3/2}+C

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integral of (x^2+x+4)sqrt(x+3)

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

\frac{2}{7} (x + 3)^{\frac{7}{2}} - 2 (x + 3)^{\frac{5}{2}} + \frac{20}{3} (x + 3)^{\frac{3}{2}} + C

Solution steps

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

Expand

(x^{2} + x + 4) x + 3 : x^{2} x + 3 + x x + 3 + 4 x + 3

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

\int x^{2} x + 3 d x = \frac{2}{7} (x + 3)^{\frac{7}{2}} - \frac{12}{5} (x + 3)^{\frac{5}{2}} + 6 (x + 3)^{\frac{3}{2}}

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

\int 4 x + 3 d x = \frac{8}{3} (x + 3)^{\frac{3}{2}}

= \frac{2}{7} (x + 3)^{\frac{7}{2}} - \frac{12}{5} (x + 3)^{\frac{5}{2}} + 6 (x + 3)^{\frac{3}{2}} + \frac{2}{5} (x + 3)^{\frac{5}{2}} - 2 (x + 3)^{\frac{3}{2}} + \frac{8}{3} (x + 3)^{\frac{3}{2}}

Simplify

\frac{2}{7} (x + 3)^{\frac{7}{2}} - \frac{12}{5} (x + 3)^{\frac{5}{2}} + 6 (x + 3)^{\frac{3}{2}} + \frac{2}{5} (x + 3)^{\frac{5}{2}} - 2 (x + 3)^{\frac{3}{2}} + \frac{8}{3} (x + 3)^{\frac{3}{2}} : \frac{2}{7} (x + 3)^{\frac{7}{2}} - 2 (x + 3)^{\frac{5}{2}} + \frac{20}{3} (x + 3)^{\frac{3}{2}}

= \frac{2}{7} (x + 3)^{\frac{7}{2}} - 2 (x + 3)^{\frac{5}{2}} + \frac{20}{3} (x + 3)^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{7} (x + 3)^{\frac{7}{2}} - 2 (x + 3)^{\frac{5}{2}} + \frac{20}{3} (x + 3)^{\frac{3}{2}} + C

What is the integral of (x^2+x+4)sqrt(x+3) ?
The integral of (x^2+x+4)sqrt(x+3) is 2/7 (x+3)^{7/2}-2(x+3)^{5/2}+20/3 (x+3)^{3/2}+C