Problem 43 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the definite integral. $$ \int_{0}^{1} e^{2 x} \sqrt{e^{2 x}+1} d x $$

Step-by-Step Solution

Answer

The value of the given definite integral is \(2\sqrt{e^{2}+1} - \frac{2}{3}(e^{2}+1)^{3/2} - 2\sqrt{2} + \frac{4}{3}\).

Step 1: Choose an appropriate substitution

Inspecting the integrand, there is a clear candidate for substitution: \(u=e^{2x}+1\). This choice will considerably simplify the square root in the integrand, and its differential \(du= 2e^{2x} dx\) will cancel out the \(e^{2x}\) in the integrand as well.

Step 2: Change the boundaries of the integral

Substituting the limits of integration, at \(x=0\), \(u=e^{2*0} +1=2\), and at \(x=1\), \(u=e^{2*1} +1=e^2+1\). This gives new limits of integration \([2, e^2+1]\).

Step 3: Rewrite the integral in terms of u

We can now rewrite our integral in terms of \(u\). Keep in mind that every \(dx\) becomes \(du/2e^{2x}=\frac{du}{2u-1}\). Therefore, the integral becomes, \(\int_{2}^{e^{2}+1} \sqrt{u} \frac{du}{2u-1}\). The \(2u-1\) can be moved out of the fraction that is under the integral, giving us \(\frac{1}{2}\int_{2}^{e^{2}+1} \frac{\sqrt{u}}{u-1} du\).

Step 4: Evaluate the integral

The integrand \(\frac{\sqrt{u}}{u-1}\) can be simplified to \(u^{-1/2} - u^{-3/2}\). Therefore, the antiderivative between 2 and \(e^2+1\) can be calculated. This antiderivative is \(2u^{1/2} - \frac{2}{3} u^{1/2}\). Substituting the upper and lower limits of the integral yields the final result.

Key Concepts

Substitution Technique

Substitution is a method that helps simplify complex integrals by changing variables. The goal is to make the integral easier to solve. In our exercise, the expression inside the square root, \(e^{2x}+1\), suggests a straightforward substitution. By letting \(u = e^{2x} + 1\), we transform the integrand.
Here’s how you can use it effectively:

• Identify a part of the integrand that complicates the integration. This might include a function inside a square root, logarithm, or exponential function.
• Choose a substitution that simplifies this expression, such as \(u = f(x)\).
• Derive \(du\) in terms of \(dx\), which is needed for changing the differential as well. In our case, \(du = 2e^{2x} dx\).

This change of variables not only simplifies the integrand but also helps in handling the differential \(dx\) within the integral.

Integration Bounds

When using substitution, converting the integration bounds is crucial to maintaining the accuracy of the integral.
Upon substituting \(u = e^{2x}+1\), the original variable, \(x\), has limits of 0 and 1. It's necessary to derive new limits in terms of \(u\).

• Start with the lower bound: When \(x=0\), substituting gives \(u=2\).
• Next, consider the upper bound: When \(x=1\), substituting gives \(u=e^2+1\).

Essentially, the substitution translates the original limits into new ones that mirror the altered variable, ensuring the definite integral evaluates correctly over the new interval \([2, e^2+1]\). Always remember to adjust the limits when you perform the substitution.

Antiderivative Calculation

The calculation of the antiderivative is pivotal in obtaining the solution to a definite integral. Once the integral's variable is substituted and the new bounds set, the next step is to find its antiderivative.
In the reformulated integral, \(\int \frac{\sqrt{u}}{u-1} du\), simplifying and finding the antiderivative involves expressing the integrand in a more workable form. Here, it transforms into: \(u^{-1/2} - u^{-3/2}\).

• Find the antiderivative of each term separately.
• For \(u^{-1/2}\), the antiderivative is \(2u^{1/2}\).
• For \(u^{-3/2}\), it is \(-\frac{2}{3}u^{-1/2}\).

After finding the antiderivative, substitute the upper and lower limits. The result is the evaluation of the definite integral between \(2\) and \(e^2+1\), leading to the problem's final solution. Understanding antiderivative calculation allows you to effectively solve definite integrals.