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 integral as the difference of a product evaluated at the endpoints and a remaining definite integral that is often simpler to compute.

A representative example is the definite integral of the inverse tangent function. Since there is no direct integration formula for arctan ⁡x, the integrand is rewritten as a product of arctan⁡ x and the constant function 1. The inverse tangent is chosen for differentiation, while the constant is integrated, simplifying the structure of the problem.

Substituting these choices into the integration by parts formula allows the first term to be evaluated directly by computing the product at the interval’s endpoints. The remaining term is a definite integral that can be solved using substitution. By letting a new variable equal 1+x², the integral is transformed into a reciprocal form that integrates to a logarithmic expression, with the limits adjusted accordingly.

Evaluating the logarithmic term at the bounds simplifies the result, since the natural logarithm of one is zero. The final expression represents the area under the inverse tangent curve between the given limits, illustrating how integration by parts can be applied effectively to definite integrals involving functions without elementary antiderivatives.