What is the value of the integral ∫(1 to 2) (x^2 + 2x)dx?

What is the value of the integral ∫(1 to 2) (x^2 + 2x)dx?

Step 1: Identify the function to integrate, which is f(x) = x^2 + 2x.
Step 2: Find the antiderivative of f(x). The antiderivative of x^2 is (1/3)x^3, and the antiderivative of 2x is x^2. So, the antiderivative F(x) = (1/3)x^3 + x^2.
Step 3: Evaluate the antiderivative at the upper limit (x = 2). F(2) = (1/3)(2^3) + (2^2) = (1/3)(8) + 4 = 8/3 + 4 = 8/3 + 12/3 = 20/3.
Step 4: Evaluate the antiderivative at the lower limit (x = 1). F(1) = (1/3)(1^3) + (1^2) = (1/3)(1) + 1 = 1/3 + 1 = 1/3 + 3/3 = 4/3.
Step 5: Subtract the value at the lower limit from the value at the upper limit: F(2) - F(1) = (20/3) - (4/3) = (20 - 4)/3 = 16/3.
Step 6: The final answer is the value of the integral, which is 16/3.

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