6.2
Integration by Parts: Definite Integrals
Definite integrals involving the product of two functions over a fixed interval can be evaluated using integration by parts. This method rewrites the…
Definite integrals of products of two functions over fixed intervals can be solved using integration by parts.
To solve the right-hand side of this expression, the difference of the product of functions will be evaluated between the interval’s endpoints, and the remaining term will be treated as a definite integral.
A useful example is the integral of the inverse tangent function. Because no standard integral formula exists for this function, the integrand is instead treated as a product of the inverse tangent and the constant 1.
The inverse tangent is taken as the function to be differentiated, and the constant is integrated.
Substituting into the integration by parts formula, the first term can be solved by directly evaluating the product at the endpoints. The remaining integral can then be solved by substitution.
A new variable, t, is set equal to 1 plus x squared and the limits of integration are adjusted. The integral then becomes a reciprocal expression that simplifies to a logarithmic form.
The natural log of one equals zero, so the final expression simplifies to show the area under the curve between the two limits.
View the full transcript and gain access to JoVE Core videos
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.