Integrating term by term, ∫3x^2dx = x^3 and ∫2dx = 2x. Thus, ∫(3x^2 + 2)dx = x^3 + 2x + C.
Questions & Step-by-step Solutions
1 item
Q
Q: Evaluate the integral ∫(3x^2 + 2)dx. (2022)
Solution: Integrating term by term, ∫3x^2dx = x^3 and ∫2dx = 2x. Thus, ∫(3x^2 + 2)dx = x^3 + 2x + C.
Steps: 6
Step 1: Identify the integral you need to evaluate, which is ∫(3x^2 + 2)dx.
Step 2: Break the integral into two separate parts: ∫3x^2dx and ∫2dx.
Step 3: For the first part, ∫3x^2dx, use the power rule of integration. The power rule states that ∫x^n dx = (x^(n+1))/(n+1) + C. Here, n is 2, so you get ∫3x^2dx = 3 * (x^(2+1))/(2+1) = 3 * (x^3)/3 = x^3.
Step 4: For the second part, ∫2dx, remember that the integral of a constant 'a' is simply 'a*x'. So, ∫2dx = 2x.
Step 5: Combine the results from Step 3 and Step 4. You have x^3 from the first part and 2x from the second part.
Step 6: Don't forget to add the constant of integration 'C' at the end. So, the final answer is x^3 + 2x + C.
