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What is the value of the integral ∫_0^1 (3x^2 + 2x) dx?

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Question: What is the value of the integral ∫_0^1 (3x^2 + 2x) dx?

Options:

  1. 1
  2. 2
  3. 3
  4. 4

Correct Answer: 2

Solution: 

Evaluating the integral gives [x^3 + x^2]_0^1 = (1 + 1) - (0 + 0) = 2.

What is the value of the integral ∫_0^1 (3x^2 + 2x) dx?

Practice Questions

Q1
What is the value of the integral ∫_0^1 (3x^2 + 2x) dx?
  1. 1
  2. 2
  3. 3
  4. 4

Questions & Step-by-Step Solutions

What is the value of the integral ∫_0^1 (3x^2 + 2x) dx?
Correct Answer: 2
  • Step 1: Identify the integral to evaluate, which is ∫_0^1 (3x^2 + 2x) dx.
  • Step 2: Find the antiderivative of the function 3x^2 + 2x. The antiderivative is x^3 + x^2.
  • Step 3: Use the Fundamental Theorem of Calculus to evaluate the antiderivative from 0 to 1. This means we will calculate [x^3 + x^2] from 0 to 1.
  • Step 4: Substitute the upper limit (1) into the antiderivative: (1^3 + 1^2) = 1 + 1 = 2.
  • Step 5: Substitute the lower limit (0) into the antiderivative: (0^3 + 0^2) = 0 + 0 = 0.
  • Step 6: Subtract the result from the lower limit from the result from the upper limit: 2 - 0 = 2.
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