Question

Difficulty: EasyDefinite Integrals and Area Under Curves

What is the value of the definite integral $\int_{0}^{2} (3x^2 + 2) \, dx$ ?

Answer: 12

To evaluate

\int_{0}^{2} (3x^2 + 2) \, dx

, integrate

3x^2 + 2

to get

x^3 + 2x

. Substituting the limits gives

(2^3 + 2(2)) - (0^3 + 2(0)) = (8 + 4) - 0 = 12

Integrate the function term by term

\int (3x^2 + 2) \, dx = x^3 + 2x

By the power rule of integration, \int 3x^2 \, dx = x^3 and \int 2 \, dx = 2x.

Apply the limits of integration from 0 to 2

[x^3 + 2x]_0^2 = (2^3 + 2(2)) - (0^3 + 2(0)) = 12 - 0 = 12

Substitute the upper limit 2 into the antiderivative and subtract the value obtained by substituting the lower limit 0.

Evaluation of Definite Integrals

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