evaluate the following integral. ∫4x(x⁴ + 9) dx ∫4x(x⁴ + 9) dx = (type an exact answer.)

Explanation:

Step1: Expand the integrand

$4x(x^{4}+9)=4x^{5}+36x$

Step2: Integrate term - by - term

$\int(4x^{5}+36x)dx=\int4x^{5}dx+\int36xdx$ Using the power rule for integration $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$), we have: $\int4x^{5}dx=4\times\frac{x^{6}}{6}=\frac{2}{3}x^{6}$ and $\int36xdx=36\times\frac{x^{2}}{2}=18x^{2}$

Step3: Combine the results

$\int4x(x^{4}+9)dx=\frac{2}{3}x^{6}+18x^{2}+C$

Answer:

$\frac{2}{3}x^{6}+18x^{2}+C$