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Explain the difference between an indefinite and a definite integral, using specific examples for each.

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Answer to a math question Explain the difference between an indefinite and a definite integral, using specific examples for each.

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1. Consideremos una función simple, por ejemplo \( f(x) = 2x \).
2. Para la integral indefinida, calculamos su antiderivada:

\int 2x \, dx = x^2 + C

El resultado es la familia de funciones parametrizadas por \( C \).

3. Para la integral definida, computamos el área entre \( x = 1 \) y \( x = 3 \):

\int_{1}^{3} 2x \, dx = [x^2]_{1}^{3} = 3^2 - 1^2 = 8

Por lo tanto, el área bajo la curva entre estos dos puntos es igual a 8.

Respuesta:
Integral indefinida: \int 2x \, dx = x^2 + C
Integral definida: \int_{1}^{3} 2x \, dx = 8

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