Solve ∫ 4r^4(27+r^5)dr

$$\int 4 r ^ { 4 } ( 27 + r ^ { 5 } ) d r$$

Evaluate

$$\frac{2r^{10}}{5}+\frac{108r^{5}}{5}+С$$

Solution Steps

Use the distributive property to multiply 4r^{4} by 27+r^{5}.

$$\int 108r^{4}+4r^{9}\mathrm{d}r$$

Integrate the sum term by term.

$$\int 108r^{4}\mathrm{d}r+\int 4r^{9}\mathrm{d}r$$

Factor out the constant in each of the terms.

$$108\int r^{4}\mathrm{d}r+4\int r^{9}\mathrm{d}r$$

Since \int r^{k}\mathrm{d}r=\frac{r^{k+1}}{k+1} for k\neq -1, replace \int r^{4}\mathrm{d}r with \frac{r^{5}}{5}. Multiply 108 times \frac{r^{5}}{5}.

$$\frac{108r^{5}}{5}+4\int r^{9}\mathrm{d}r$$

Since \int r^{k}\mathrm{d}r=\frac{r^{k+1}}{k+1} for k\neq -1, replace \int r^{9}\mathrm{d}r with \frac{r^{10}}{10}. Multiply 4 times \frac{r^{10}}{10}.

$$\frac{108r^{5}+2r^{10}}{5}$$

If F\left(r\right) is an antiderivative of f\left(r\right), then the set of all antiderivatives of f\left(r\right) is given by F\left(r\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.

$$\frac{108r^{5}}{5}+\frac{2r^{10}}{5}+С$$

Differentiate w.r.t. r

$$4r^{4}\left(r^{5}+27\right)$$