What is the integral of (x^2+x+9)sqrt(x+8) ?

The integral of (x^2+x+9)sqrt(x+8) is 2/7 (x+8)^{7/2}-6(x+8)^{5/2}+130/3 (x+8)^{3/2}+C

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integral of (x^2+x+9)sqrt(x+8)

\int (x^{2} + x + 9) x + 8 d x

\frac{2}{7} (x + 8)^{\frac{7}{2}} - 6 (x + 8)^{\frac{5}{2}} + \frac{130}{3} (x + 8)^{\frac{3}{2}} + C

Solution steps

\int (x^{2} + x + 9) x + 8 d x

Expand

(x^{2} + x + 9) x + 8 : x^{2} x + 8 + x x + 8 + 9 x + 8

= \int x^{2} x + 8 + x x + 8 + 9 x + 8 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 + 8 d x + \int x x + 8 d x + \int 9 x + 8 d x

\int x^{2} x + 8 d x = \frac{2}{7} (x + 8)^{\frac{7}{2}} - \frac{32}{5} (x + 8)^{\frac{5}{2}} + \frac{128}{3} (x + 8)^{\frac{3}{2}}

\int x x + 8 d x = \frac{2}{5} (x + 8)^{\frac{5}{2}} - \frac{16}{3} (x + 8)^{\frac{3}{2}}

\int 9 x + 8 d x = 6 (x + 8)^{\frac{3}{2}}

= \frac{2}{7} (x + 8)^{\frac{7}{2}} - \frac{32}{5} (x + 8)^{\frac{5}{2}} + \frac{128}{3} (x + 8)^{\frac{3}{2}} + \frac{2}{5} (x + 8)^{\frac{5}{2}} - \frac{16}{3} (x + 8)^{\frac{3}{2}} + 6 (x + 8)^{\frac{3}{2}}

Simplify

\frac{2}{7} (x + 8)^{\frac{7}{2}} - \frac{32}{5} (x + 8)^{\frac{5}{2}} + \frac{128}{3} (x + 8)^{\frac{3}{2}} + \frac{2}{5} (x + 8)^{\frac{5}{2}} - \frac{16}{3} (x + 8)^{\frac{3}{2}} + 6 (x + 8)^{\frac{3}{2}} : \frac{2}{7} (x + 8)^{\frac{7}{2}} - 6 (x + 8)^{\frac{5}{2}} + \frac{130}{3} (x + 8)^{\frac{3}{2}}

= \frac{2}{7} (x + 8)^{\frac{7}{2}} - 6 (x + 8)^{\frac{5}{2}} + \frac{130}{3} (x + 8)^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{7} (x + 8)^{\frac{7}{2}} - 6 (x + 8)^{\frac{5}{2}} + \frac{130}{3} (x + 8)^{\frac{3}{2}} + C

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What is the integral of (x^2+x+9)sqrt(x+8) ?
The integral of (x^2+x+9)sqrt(x+8) is 2/7 (x+8)^{7/2}-6(x+8)^{5/2}+130/3 (x+8)^{3/2}+C