What is the integral of (r^3+5)^2 ?

The integral of (r^3+5)^2 is (r^7)/7+(5r^4)/2+25r+C

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integral of (r^3+5)^2

\int (r^{3} + 5)^{2} d r

\frac{r ^{7}}{7} + \frac{5 r ^{4}}{2} + 25 r + C

Solution steps

\int (r^{3} + 5)^{2} d r

Expand

(r^{3} + 5)^{2} : r^{6} + 10 r^{3} + 25

= \int r^{6} + 10 r^{3} + 25 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^{6} d r + \int 10 r^{3} d r + \int 25 d r

\int r^{6} d r = \frac{r ^{7}}{7}

\int 10 r^{3} d r = \frac{5 r ^{4}}{2}

\int 25 d r = 25 r

= \frac{r ^{7}}{7} + \frac{5 r ^{4}}{2} + 25 r

Add a constant to the solution

= \frac{r ^{7}}{7} + \frac{5 r ^{4}}{2} + 25 r + C

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What is the integral of (r^3+5)^2 ?
The integral of (r^3+5)^2 is (r^7)/7+(5r^4)/2+25r+C