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

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

Options:

Correct Answer: 8

Solution:

∫(2x + 3)dx = [x^2 + 3x] from 1 to 2 = (4 + 6) - (1 + 3) = 10 - 4 = 6.

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

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

Step 1: Identify the integral you need to evaluate: ∫(1 to 2) (2x + 3)dx.
Step 2: Find the antiderivative of the function (2x + 3). The antiderivative is x^2 + 3x.
Step 3: Write the antiderivative with limits of integration: [x^2 + 3x] from 1 to 2.
Step 4: Substitute the upper limit (2) into the antiderivative: (2^2 + 3*2) = (4 + 6) = 10.
Step 5: Substitute the lower limit (1) into the antiderivative: (1^2 + 3*1) = (1 + 3) = 4.
Step 6: Subtract the result of the lower limit from the upper limit: 10 - 4 = 6.
Step 7: The value of the integral is 6.

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.
Evaluation of Limits – Substituting the upper and lower limits into the antiderivative to find the net area.