What is the integral of r^2(r^3+6)^3 ?

The integral of r^2(r^3+6)^3 is (r^{12})/(12)+2r^9+18r^6+72r^3+C

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integral of r^2(r^3+6)^3

\int r^{2} (r^{3} + 6)^{3} d r

\frac{r ^{12}}{12} + 2 r^{9} + 18 r^{6} + 72 r^{3} + C

Solution steps

\int r^{2} (r^{3} + 6)^{3} d r

Expand

r^{2} (r^{3} + 6)^{3} : r^{11} + 18 r^{8} + 108 r^{5} + 216 r^{2}

= \int r^{11} + 18 r^{8} + 108 r^{5} + 216 r^{2} d r

Apply the Sum Rule:

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

= \int r^{11} d r + \int 18 r^{8} d r + \int 108 r^{5} d r + \int 216 r^{2} d r

\int r^{11} d r = \frac{r ^{12}}{12}

\int 18 r^{8} d r = 2 r^{9}

\int 108 r^{5} d r = 18 r^{6}

\int 216 r^{2} d r = 72 r^{3}

= \frac{r ^{12}}{12} + 2 r^{9} + 18 r^{6} + 72 r^{3}

Add a constant to the solution

= \frac{r ^{12}}{12} + 2 r^{9} + 18 r^{6} + 72 r^{3} + C

\frac{\partial}{\partial x} (cos^{9} (x^{7} y^{9}))

y = 7 arctan (x + 1 + x^{2})

y = 5 x, y = \frac{5}{2} x, y = 84 - x^{2}

x \to 8 lim (ln (x - 8))

What is the integral of r^2(r^3+6)^3 ?
The integral of r^2(r^3+6)^3 is (r^{12})/(12)+2r^9+18r^6+72r^3+C