Evaluate the indefinite integral: $\int (9x^2+81)dx$

• A. $3x^3+81x+C$
• B. $3x^2+81x+C$
• C. $9x^2+81x+C$
• D. $9x^3+81x^2+C$

Answer: (A)

Verified step by step guidance

1. Step 1: Recall the formula for the indefinite integral of a polynomial term. For a term of the form x^n, the integral is \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where C is the constant of integration.
2. Step 2: Break the given integral \( \int (9x^2 + 81) \, dx \) into separate terms: \( \int 9x^2 \, dx + \int 81 \, dx \). This allows us to integrate each term individually.
3. Step 3: For the first term \( \int 9x^2 \, dx \), factor out the constant 9 and apply the power rule: \( 9 \int x^2 \, dx = 9 \cdot \frac{x^{2+1}}{2+1} = 9 \cdot \frac{x^3}{3} \).
4. Step 4: For the second term \( \int 81 \, dx \), recall that the integral of a constant is the constant multiplied by x: \( \int 81 \, dx = 81x \).
5. Step 5: Combine the results from Step 3 and Step 4, and add the constant of integration C: \( 3x^3 + 81x + C \). This is the general solution for the indefinite integral.