6.2
Integración por partes: Integrales definidas
Las integrales definidas que implican el producto de dos funciones en un intervalo fijo se pueden evaluar mediante la integración por partes. Este mét…
Las integrales definidas de productos de dos funciones sobre intervalos fijos pueden resolverse mediante integración por partes.
Para resolver el lado derecho de esta expresión, se evaluará la diferencia del producto de funciones entre los extremos del intervalo, y el término restante se tratará como una integral definida.
Un ejemplo útil es la integral de la función tangente inversa. Como no existe una fórmula integral estándar para esta función, el integrando se trata en cambio como un producto de la tangente inversa y la constante 1.
La tangente inversa se toma como la función a diferenciar y la constante se integra.
Sustituyendo en la fórmula de integración por partes, el primer término puede resolverse evaluando directamente el producto en los puntos finales. La integral restante puede resolverse entonces por sustitución.
Una nueva variable, t, se establece igual a 1 más x al cuadrado y se ajustan los límites de integración. La integral entonces se convierte en una expresión recíproca que se simplifica a una forma logarítmica.
El logaritmo natural de uno es igual a cero, por lo que la expresión final se simplifica para mostrar el área bajo la curva entre los dos límites.
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Q1: How does integration by parts apply to definite integrals?
Integration by parts for definite integrals rewrites the integral as the difference of a product evaluated at the endpoints minus a remaining definite integral. The formula transforms products of two functions into simpler forms by choosing which function to differentiate and which to integrate, then evaluating the product term at the interval's boundaries.
Q2: Why is the inverse tangent function difficult to integrate directly?
The inverse tangent function has no standard integration formula, so it cannot be integrated using elementary antiderivative techniques. To solve integrals involving arctan(x), the integrand is rewritten as a product of the inverse tangent and the constant 1, allowing integration by parts to be applied effectively.
Q3: What role does substitution play after applying integration by parts?
After integration by parts simplifies the original integral, substitution is used to evaluate the remaining definite integral. By introducing a new variable such as t = 1 + x², the integral is transformed into a reciprocal form that integrates to a logarithmic expression, with limits adjusted accordingly for the new variable.
Q4: How do you evaluate the product term in the integration by parts formula?
The product term is evaluated by computing the product of the two chosen functions at each endpoint of the integration interval, then finding the difference between these values. This direct evaluation eliminates the need to find an antiderivative for the product itself.
Q5: Why does the natural logarithm of one equal zero in the final answer?
The natural logarithm of one equals zero by definition, since e^0 = 1. When evaluating logarithmic expressions at the bounds of a definite integral, this property simplifies the result, often eliminating terms and leaving a cleaner final expression for the area under the curve.
Q6: What does the final result of integrating inverse tangent represent?
The final result represents the area under the inverse tangent curve between the given integration limits. This geometric interpretation shows how integration by parts successfully evaluates definite integrals of functions without elementary antiderivatives, providing both a numerical answer and conceptual understanding.
Q7: How does integration by parts for definite integrals differ from integration by parts for indefinite integrals?
Definite integrals include fixed endpoints that are substituted directly into the product term, eliminating the constant of integration. With indefinite integrals, the constant of integration remains in the final answer. Both methods use the same formula structure, but definite integrals yield numerical results while indefinite integrals produce families of antiderivatives.