Calculate the integral ∫ (x^2 + 2x + 1) dx.

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Question: Calculate the integral ∫ (x^2 + 2x + 1) dx.

Options:

Correct Answer: (1/3)x^3 + x^2 + x + C

Solution:

The integral of x^2 is (1/3)x^3, the integral of 2x is x^2, and the integral of 1 is x. Thus, ∫ (x^2 + 2x + 1) dx = (1/3)x^3 + x^2 + x + C.

Calculate the integral ∫ (x^2 + 2x + 1) dx.

Calculate the integral ∫ (x^2 + 2x + 1) dx.

Step 1: Identify the function to integrate, which is (x^2 + 2x + 1).
Step 2: Break down the integral into parts: ∫(x^2) dx, ∫(2x) dx, and ∫(1) dx.
Step 3: Calculate the integral of x^2. The formula is ∫(x^n) dx = (1/(n+1))x^(n+1). Here, n=2, so ∫(x^2) dx = (1/3)x^3.
Step 4: Calculate the integral of 2x. Using the same formula, n=1, so ∫(2x) dx = 2 * (1/2)x^2 = x^2.
Step 5: Calculate the integral of 1. This is simply ∫(1) dx = x.
Step 6: Combine all the results from Steps 3, 4, and 5: (1/3)x^3 + x^2 + x.
Step 7: Add the constant of integration, C, to the final result: ∫(x^2 + 2x + 1) dx = (1/3)x^3 + x^2 + x + C.

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