What is the integral of (x^2+x+9)sqrt(x+7) ?

The integral of (x^2+x+9)sqrt(x+7) is 2/7 (x+7)^{7/2}-26/5 (x+7)^{5/2}+34(x+7)^{3/2}+C

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integral of (x^2+x+9)sqrt(x+7)

\int (x^{2} + x + 9) x + 7 d x

\frac{2}{7} (x + 7)^{\frac{7}{2}} - \frac{26}{5} (x + 7)^{\frac{5}{2}} + 34 (x + 7)^{\frac{3}{2}} + C

Solution steps

\int (x^{2} + x + 9) x + 7 d x

Expand

(x^{2} + x + 9) x + 7 : x^{2} x + 7 + x x + 7 + 9 x + 7

= \int x^{2} x + 7 + x x + 7 + 9 x + 7 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 + 7 d x + \int x x + 7 d x + \int 9 x + 7 d x

\int x^{2} x + 7 d x = \frac{2}{7} (x + 7)^{\frac{7}{2}} - \frac{28}{5} (x + 7)^{\frac{5}{2}} + \frac{98}{3} (x + 7)^{\frac{3}{2}}

\int x x + 7 d x = \frac{2}{5} (x + 7)^{\frac{5}{2}} - \frac{14}{3} (x + 7)^{\frac{3}{2}}

\int 9 x + 7 d x = 6 (x + 7)^{\frac{3}{2}}

= \frac{2}{7} (x + 7)^{\frac{7}{2}} - \frac{28}{5} (x + 7)^{\frac{5}{2}} + \frac{98}{3} (x + 7)^{\frac{3}{2}} + \frac{2}{5} (x + 7)^{\frac{5}{2}} - \frac{14}{3} (x + 7)^{\frac{3}{2}} + 6 (x + 7)^{\frac{3}{2}}

Simplify

\frac{2}{7} (x + 7)^{\frac{7}{2}} - \frac{28}{5} (x + 7)^{\frac{5}{2}} + \frac{98}{3} (x + 7)^{\frac{3}{2}} + \frac{2}{5} (x + 7)^{\frac{5}{2}} - \frac{14}{3} (x + 7)^{\frac{3}{2}} + 6 (x + 7)^{\frac{3}{2}} : \frac{2}{7} (x + 7)^{\frac{7}{2}} - \frac{26}{5} (x + 7)^{\frac{5}{2}} + 34 (x + 7)^{\frac{3}{2}}

= \frac{2}{7} (x + 7)^{\frac{7}{2}} - \frac{26}{5} (x + 7)^{\frac{5}{2}} + 34 (x + 7)^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{7} (x + 7)^{\frac{7}{2}} - \frac{26}{5} (x + 7)^{\frac{5}{2}} + 34 (x + 7)^{\frac{3}{2}} + C

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What is the integral of (x^2+x+9)sqrt(x+7) ?
The integral of (x^2+x+9)sqrt(x+7) is 2/7 (x+7)^{7/2}-26/5 (x+7)^{5/2}+34(x+7)^{3/2}+C