Integration by Parts of Definite Integrals: Ignoring Limits Temporarily

When performing integration by parts on a definite integral, we often temporarily ignore the upper and lower limits of the integral. This might seem counterintuitive, but it simplifies the process and focuses us on the core concept. This article will explore why this ignoring of limits is done, the steps involved, and the importance of evaluating the boundary terms at the end.

Why We Temporarily Ignore Limits

The primary reason for ignoring the limits during the integration by parts process is to ensure that the algebraic manipulation of the integrand is clear. By focusing on the form of the integral without the limits, we can better structure the problem and ensure that all parts of the integral are correctly integrated and manipulated.

Focusing on the Form

Let's consider the integration by parts formula:

uv - ∫v du

When we choose u and dv, we are essentially breaking down the integrand into more manageable parts. This process is purely algebraic and does not yet require the definition of the limits. Ignoring the limits helps us concentrate on the structure and composition of the integral before we deal with the specific bounds.

Steps in Integration by Parts with Definite Integrals

The process of integration by parts on a definite integral involves several steps:

Choose u and dv

The first step is to identify parts of the integrand to set as u and dv. This often requires experience and intuition to make the best choice. The goal is to set u so that its derivative becomes easier to integrate, and dv so that its integral is simpler to compute.

Differentiate and Integrate

Once we have chosen u and dv, we need to compute du and v. This involves differentiating u and integrating dv. These steps are crucial for applying the integration by parts formula correctly.

Apply the Formula

After identifying u, dv, du, and v, we substitute them into the integration by parts formula:

uv - ∫v du

This formula transforms the original integral into a new form that might be easier to solve, especially if one of the terms is simpler to integrate than the original integrand.

Evaluate the Boundary Terms

The term uv - ∫v du consists of two main parts: the product of u and v evaluated at the upper and lower limits, and the remaining integral. The product of u and v at the boundaries is essential and cannot be ignored; it is often referred to as the boundary terms.

For example, when evaluating:

∫01 x arctan x dx

We ignore the limits initially:

[∫01 x arctan x dx
[01 - ∫01 01 [1 - 01]

This example demonstrates the importance of evaluating the boundary terms at the end of the integration process.

Final Evaluation

The final step in evaluating a definite integral is to return to the limits of integration. This ensures that the process is methodical and free of errors. Even though we temporarily ignore the limits during the intermediate steps, we must remember to include them in the final calculations to obtain the correct value of the integral.

Conclusion

In conclusion, while we may momentarily disregard the limits during the integration by parts process, they are crucial for the final evaluation of the definite integral. Always ensure that the boundary terms are evaluated correctly to obtain the accurate result. Ignoring the limits simplifies the process and helps maintain clarity, but including them at the end is essential for accuracy.

Related Keywords

Integration by parts, definite integrals, boundary terms