Problem 40 - Calculus Early Transcendentals

Question

Use a change of variables to evaluate the following definite integrals. $$\int_{0}^{2} \frac{2 x}{\left(x^{2}+1\right)^{2}} d x$$

Step-by-Step Solution

Answer

Question: Evaluate the definite integral using substitution: $$\int_{0}^{2} \frac{2 x}{\left(x^{2}+1\right)^{2}} d x$$ Answer: The value of the integral is $$\frac{4}{5}$$

Step 1: Choose a substitution

Let \(u = x^2 + 1\). Then the differential \(du\) can be found by taking the derivative of \(u\) with respect to \(x\) and then multiplying by \(dx\). We have \(du = 2x\, dx\). Now we can substitute \(u\) and \(du\) into our integral.

Step 2: Change the limits of integration

When we change variables, we must also change the limits of integration. For the given integral, our original limits are \(x = 0\) and \(x = 2\). Using our substitution, we need to find the new limits for \(u\): For the lower limit, when \(x = 0\), we have \(u = (0)^2 + 1 = 1\). For the upper limit, when \(x = 2\), we have \(u = (2)^2 + 1 = 5\). The new limits for the integral in terms of \(u\) will be \(1\) and \(5\).

Step 3: Substitute the new variables

Now, we can substitute \(u\) and \(du\) into the integral and replace the limits of integration. We get: $$\int_{1}^{5} \frac{1}{u^2} du$$

Step 4: Evaluate the integral

Now, we can evaluate the integral. The antiderivative of \(\frac{1}{u^2}\) is \(-\frac{1}{u}\): $$\left[-\frac{1}{u}\right]_{1}^{5} = -\frac{1}{5} - (-\frac{1}{1}) = -\frac{1}{5} + 1 = \frac{4}{5}$$ Therefore, we find that the value of the integral is \(\frac{4}{5}\).

Key Concepts

Change of Variables

In calculus, the change of variables technique is a strategic way to simplify complicated integrals. The essence of this method lies in replacing a complex expression with a simpler one by introducing a new variable.
This new variable, often denoted by a different letter, typically corresponds to an expression in the original variable that appears frequently in the integrand.

• This change facilitates simplification by potentially making the integrand easier to integrate.
• It rearranges the integral into a form that may be more straightforward to evaluate.

During this process, you also replace the differential of the variable, such as changing from \(dx\) to \(du\). In the given exercise, substituting \(u = x^2 + 1\) simplifies the expression \(\frac{2x}{(x^2 + 1)^2}\) in terms of \(u\). The goal of changing variables is to streamline the integral into a more manageable form.

Limits of Integration

When applying the change of variables, adjusting the limits of integration is crucial. These limits define the boundaries over which the integration occurs.
In definite integrals, changing variables impacts these boundaries, which means you'll need to calculate new limits in terms of the new variable.To find these new limits:

• Substitute the original limit values into the expression for the new variable.
• For example, in the integral \( \int_{0}^{2} \frac{2x}{(x^2+1)^2} dx \) with \( u = x^2 + 1 \):
  • When \( x = 0 \), \( u = (0)^2 + 1 = 1 \).
  • When \( x = 2 \), \( u = (2)^2 + 1 = 5 \).

These calculations yield the new integration limits of \(1\) to \(5\). Correctly determining these limits is essential to ensure the integration covers the intended range.

Substitution Method

The substitution method is a primary strategy within calculus for evaluating integrals. It's particularly helpful when you encounter a complex integrand that can be made simpler through substitution.
Here’s how it typically works:

• Select a substitution for part of the integral, often denoting it with \(u\), as in \(u = x^2 + 1\).
• Compute the differential of \(u\) and substitute back into the integral, transforming both the integrand and the differential. For the exercise in question, this involved converting \(du = 2x \, dx\).

Once the integral is entirely in terms of \(u\), it often becomes simpler to evaluate. In our example, this approach converts \(\int_{0}^{2} \frac{2x}{(x^2+1)^2} \) to \(\int_{1}^{5} \frac{1}{u^2} du\), which is straightforward to integrate. After finding the antiderivative, you can evaluate it using the new limits to solve the integral.