evaluate the following integral. ∫9x(x³ + 8) dx ∫9x(x³ + 8) dx = □ (type an exact answer.)
evaluate the following integral. ∫9x(x³ + 8) dx ∫9x(x³ + 8) dx = □ (type an exact answer.)
Answer
Explanation:
Step1: Expand the integrand
$9x(x^{3}+8)=9x^{4}+72x$
Step2: Integrate term - by - term
$\int(9x^{4}+72x)dx=\int9x^{4}dx+\int72xdx$ Using the power rule for integration $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$), we have: $\int9x^{4}dx=9\times\frac{x^{5}}{5}=\frac{9x^{5}}{5}$ and $\int72xdx=72\times\frac{x^{2}}{2}=36x^{2}$
Step3: Combine the results
$\int9x(x^{3}+8)dx=\frac{9x^{5}}{5}+36x^{2}+C$
Answer:
$\frac{9x^{5}}{5}+36x^{2}+C$