Question

When using a change of variables \(u=g(x)\) to evaluate the definite integral \(\int_{a}^{b} f(g(x)) g^{\prime}(x) d x,\) how are the limits of integration transformed?

Short answer

Based on the given solution, the limits of integration transform as follows: Lower limit: \(u = g(a)\) Upper limit: \(u = g(b)\) When using the change of variables \(u = g(x)\) to evaluate the definite integral \(\int_a^b f(g(x)) g'(x) dx\), the limits of the integral transform, and we now integrate from \(u=g(a)\) to \(u=g(b)\).

Step-by-step solution

Define the change of variables

We are given the substitution \(u = g(x)\). Our task is to transform the integral and limits of integration with respect to this change of variables.

Express the differential in terms of dx

To make the substitution, we need to express the differential \(dx\) in terms of \(du\). Since \(u = g(x)\), we can differentiate both sides with respect to x to get: $$ \frac{d u}{d x}=g^{\prime}(x) $$ Now, rearrange the equation to solve for \(dx\): $$ d x=\frac{d u}{g^{\prime}(x)} $$

Substitute u and du into the integral

Now, we substitute \(u\) for \(g(x)\) and \(dx\) for \(\frac{du}{g'(x)}\) into the original integral: $$ \int_a^b f(g(x)) g'(x) dx = \int_{g(a)}^{g(b)} f(u) \frac{du}{g'(x)} $$

Transforming the limits of integration

Notice that the integral now has new limits of integration. Instead of integrating from \(x=a\) to \(x=b\), we are now integrating from \(u=g(a)\) to \(u=g(b)\). So, the transformed limits of integration are: Lower limit: \(u = g(a)\) Upper limit: \(u = g(b)\)

Write the final transformed integral

After transforming the limits of integration, we have the following definite integral with respect to the variable \(u\): $$ \int_{g(a)}^{g(b)} f(u) \frac{du}{g'(x)} $$ Now, we have the definite integral with transformed limits of integration, which is the desired result.

Key concepts

Definite Integrals

Definite integrals are a fundamental concept in calculus that represent the net area under a curve over a given interval. As opposed to indefinite integrals, which provide the general form of the antiderivative, a definite integral is computed between two specific bounds, usually denoted as \(a\) and \(b\), and it yields a numeric value. The notation \(\int_{a}^{b} f(x) dx\) signifies the definite integral of the function \(f(x)\) between the lower limit \(a\) and upper limit \(b\).

This concept is essential because it enables us to calculate quantities such as areas, volumes, displacement, and more in applied fields. Understanding how to work with definite integrals allows for the evaluation of more complex scenarios, especially when a direct integration approach is challenging or cumbersome.

Substitution Method

The substitution method, also known as the change of variables method, is a technique used to simplify complex integrals. In the context of definite integrals, it can transform an integral with a challenging integrand into one that is easier to evaluate. The substitution usually involves introducing a new variable, \(u\), where \(u = g(x)\) for some function \(g\) related to the integrand.

When this new variable \(u\) replaces parts of the original integrand, it can often make the resulting integral more straightforward to compute. This process effectively 'substitutes' a tricky part of the integrand with a simpler expression, which is why the method is so named. It's worth mentioning that careful attention is needed to ensure that the substitution is chosen wisely, as an inappropriate substitution might not yield any simplification.

U-Substitution

U-substitution is a specific form of the substitution method tailored for integrating functions. It is often used when an integral contains a function and its derivative within it. By letting \( u = g(x) \), where \(g(x)\) is part of the integrand, the integral can often be transformed into a simpler form in terms of \(u\), making it easier to evaluate.

The process typically involves four main steps: identifying the inner function \(g(x)\), defining \(u = g(x)\), expressing \(dx\) in terms of \(du\), and substituting \(u\) and \(du\) into the original integral. A successful u-substitution makes the integral simpler and often leads to an antiderivative that can then be evaluated with the transformed limits of integration.

Integration Limits Transformation

When applying u-substitution to a definite integral, not only must the integrand be transformed, but the integration limits must also change to reflect the new variable of integration. The original integration limits are defined in terms of \(x\), but after the substitution, these limits become irrelevant since the integration is now with respect to \(u\).

To find the new limits, one must substitute the original \(x\)-bounds into the substitution equation \(u = g(x)\). This yields the new lower limit \(u = g(a)\) and the upper limit \(u = g(b)\), where \(a\) and \(b\) are the original limits in terms of \(x\). With these new limits, the transformed integral can be accurately evaluated, ensuring that the resulting value corresponds properly to the specified interval in the \(x\)-domain.