Problem 31 - Calculus Early Transcendentals

Question

Use a table of integrals to determine the following indefinite integrals. These integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table. $$\int \frac{e^{x}}{\sqrt{e^{2 x}+4}} d x$$

Step-by-Step Solution

Answer

Question: Determine the indefinite integral of the given function: $$\int \frac{e^x}{\sqrt{e^{2x}+4}} dx$$ Answer: The indefinite integral of the given function is: $$\int \frac{e^x}{\sqrt{e^{2x}+4}} dx = \frac{1}{2}\log{(e^x\sqrt{2}+\sqrt{e^{2x}+4})}+C$$

Step 1: Find a substitution for the integrand

We will use a substitution method to make the expression easier to integrate. Since the function has both exponential and radical terms, we can try to find a substitution that will reduce these terms to a simpler, more recognizable form. Let's consider the following substitution: $$u = e^x$$ Since we're substituting, we also need to find the relation between the differentials, so differentiation of u gives: $$\frac{d u}{d x}=e^x$$ And thus, $$d x=\frac{d u}{e^x}$$ Now substitute u and dx into the integral: $$\int \frac{e^{x}}{\sqrt{e^{2 x}+4}} d x = \int \frac{1}{\sqrt{u^2+4}}\cdot u \frac{d u}{u}$$ As we can see, u cancels out, and our integral simplifies to: $$\int \frac{1}{\sqrt{u^2+4}} d u$$

Step 2: Further transformation of the integrand

Our integrand is still a little complicated, so we'll try completing the square on the expression under the square root, by introducing a new change of variables. Let's make the following substitution: $$w = u\sqrt{2}$$ Now, we find the relation between the differentials of w and u: $$d w = \sqrt{2} d u \Rightarrow d u =\frac{1}{\sqrt{2}} d w$$ Substitute w and du into the integral: $$\int \frac{1}{\sqrt{u^2+4}} d u = \int \frac{1}{\sqrt{(\frac{w}{\sqrt{2}})^2+4}}\cdot \frac{1}{\sqrt{2}} d w$$ This results in: $$\frac{1}{\sqrt{2}}\int \frac{1}{\sqrt{\frac{w^2}{2}+4}} d w$$ By adjusting the terms inside the square root, we end up with: $$\frac{1}{\sqrt{2}}\int \frac{1}{\sqrt{\frac{w^2+8}{2}}} d w = \frac{1}{2}\int \frac{d w}{\sqrt{w^2+8}}$$

Step 3: Find the integral from the table of integrals

Now it's time to refer to the table of integrals to find the indefinite integral. From the table, we have the following formula: $$\int \frac{d x}{\sqrt{x^2+a^2}} = \log{(x+\sqrt{x^2+a^2})}+C$$ Using this formula, we can rewrite our integral as: $$\frac{1}{2}\int \frac{d w}{\sqrt{w^2+8}} = \frac{1}{2} \log{(w+\sqrt{w^2+8})}+C$$

Step 4: Substitute back the original variables

Now we want to bring our result back in terms of the original variable, x. We work our way backwards with the substitutions: First, substitute w back in terms of u: $$\frac{1}{2}\log{(\frac{u\sqrt{2}+\sqrt{(\frac{u^2\sqrt{2}}{2})+8}})}+C$$ Then, substitute u back in terms of x: $$\frac{1}{2}\log{(e^x\sqrt{2}+\sqrt{e^{2x}+4})}+C$$ This is our final solution for the indefinite integral: $$\int \frac{e^{x}}{\sqrt{e^{2 x}+4}} d x = \frac{1}{2}\log{(e^x\sqrt{2}+\sqrt{e^{2x}+4})}+C$$

Key Concepts

Integration by Substitution

Integration by substitution, often referred to as u-substitution, is an indispensable technique for solving indefinite integrals, especially when dealing with complex expressions. This process involves identifying a portion of the integrand that can be simplified by substituting it with a new variable, typically denoted as 'u'. The main goal here is to transform an intricate integral into a simpler form that is more straightforward to integrate.

In the exercise provided, the substitution chosen is u = e^x. After substituting 'u' into the integral, we are immediately able to cancel out terms, simplifying our integral significantly. The basic steps are to choose a suitable 'u', differentiate to find 'du', replace all occurrences of the original variable with 'u', then integrate with respect to 'u'. Remember, it's crucial to differentiate correctly and ensure that all parts of the integrand, including 'dx', are substituted. It's equally vital to ensure that the limits of integration are adjusted if the integral is definite.

Completing the Square

Completing the square is a technique used to rewrite quadratics in a form where they can be easily integrated or solved. The method involves creating a perfect square trinomial from the quadratic expression, thus simplifying the integral's structure. It's predominantly used in integrals involving radicals and can facilitate transforms that lead to more amenable integral forms.

In the exercise example, the square completion wasn't immediately necessary, but after the initial substitution, we encounter a square root involving u^2, which prompts a further variable change. This second substitution transforms our integrand into a form that matches a known integral from our table of integrals. While completing the square might sometimes seem like an extra step, it's one that smooths the path to our final solution. Moreover, it's extremely helpful when encountering integrals involving roots of quadratic expressions or forms that resemble the derivatives of inverse trigonometric functions.

Table of Integrals

A table of integrals is an invaluable tool in calculus, serving as a ready reference to find the antiderivatives of common functions. These tables list various integral formulas that students can use to solve integrals more quickly than attempting to derive the antiderivative from first principles every time. While a firm understanding of fundamental integration techniques is essential, knowing how to use these tables effectively can save time and streamline the integration process.

The table of integrals was used in the final steps of the exercise to find the integral of a function that was already manipulated into a recognizable form. Always ensure that the integral matches exactly with a formula in the table; sometimes this requires careful manipulation of constants or another change of variables, as we have done in our example. Remember, the use of integration tables simplifies the integration process significantly and is a testament to the cumulative knowledge of mathematical techniques.

Integration Techniques

There are varied integration techniques at a mathematician's disposal, each suited to tackle different types of integrands. Aside from substitution and completing the square, as we saw in the textbook example, other methods include integration by parts, partial fractions, and trigonometric substitution. A strong grasp of these techniques allows students to approach a broad array of integrals with confidence.

Understanding when and how to apply particular methods is crucial. For instance, in our textbook example, after initially trying substitution, we recognized the need for completing the square, followed by referencing a table of integrals for the final form. Each step was a judicious decision to move closer to a workable solution. What is paramount is the adaptability to assess which method, or combination of methods, will lead to an executable integration process. Sometimes integrals require a clever insight or a methodical approach through trial and error. Developing proficiency with these techniques is a cornerstone of calculus education and helps students navigate complex integrals like the example provided.