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Evaluate the integral ∫(1 to 2) (3x^2 - 2)dx.

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Question: Evaluate the integral ∫(1 to 2) (3x^2 - 2)dx.

Options:

  1. 3
  2. 4
  3. 5
  4. 6

Correct Answer: 3

Solution: 

The integral evaluates to [(x^3 - 2x)] from 1 to 2 = (8 - 4) - (1 - 2) = 5.

Evaluate the integral ∫(1 to 2) (3x^2 - 2)dx.

Practice Questions

Q1
Evaluate the integral ∫(1 to 2) (3x^2 - 2)dx.
  1. 3
  2. 4
  3. 5
  4. 6

Questions & Step-by-Step Solutions

Evaluate the integral ∫(1 to 2) (3x^2 - 2)dx.
Correct Answer: 5
  • Step 1: Identify the integral to evaluate: ∫(1 to 2) (3x^2 - 2)dx.
  • Step 2: Find the antiderivative of the function 3x^2 - 2. The antiderivative is x^3 - 2x.
  • Step 3: Evaluate the antiderivative at the upper limit (x = 2): (2^3 - 2*2) = (8 - 4) = 4.
  • Step 4: Evaluate the antiderivative at the lower limit (x = 1): (1^3 - 2*1) = (1 - 2) = -1.
  • Step 5: Subtract the value at the lower limit from the value at the upper limit: 4 - (-1) = 4 + 1 = 5.
  • Definite Integral – The process of calculating the area under a curve defined by a function over a specific interval.
  • Fundamental Theorem of Calculus – Relates differentiation and integration, allowing the evaluation of definite integrals using antiderivatives.
  • Polynomial Integration – Involves finding the antiderivative of polynomial functions, which is a key skill in calculus.
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