Question

perform the indicated multiplication and then integrate. use c for the constant of integration (int x^5(8x + 5) , dx)

Explanation:

Step1: Distribute \( x^5 \)

Multiply \( x^5 \) with each term in \( (8x + 5) \):
\( x^5(8x + 5) = 8x^6 + 5x^5 \)

Step2: Integrate term by term

Use the power rule for integration \( \int x^n dx = \frac{x^{n+1}}{n+1} + C \) (for \( n
eq -1 \)):

• For \( 8x^6 \): \( \int 8x^6 dx = 8 \cdot \frac{x^{7}}{7} = \frac{8x^7}{7} \)
• For \( 5x^5 \): \( \int 5x^5 dx = 5 \cdot \frac{x^{6}}{6} = \frac{5x^6}{6} \)

Combine results and add \( C \).

Answer:

\( \frac{8x^7}{7} + \frac{5x^6}{6} + C \)