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Chapter 5: Problem 36

In Exercises \(33-42\) find the indefinite integral. $$\int \sec \frac{x}{2} d x$$

Short Answer

Expert verified
The integral \(\int \sec \frac{x}{2} dx = 2\ln|\sec(\frac{x}{2})+\tan(\frac{x}{2})| + C \)

Step by step solution

01

Identify the integral and substitution

Observe the integral and identify that the integration will be done in respect to \(x\). A common approach is to let \(u = \frac{x}{2}\) or \(x=2u\). And then, compute the differential \(\frac{dx}{du} = 2\).
02

Apply substitution and convert integrals

Now we substitute \(x\) with \(2u\) in the integral and then apply the change in terms of differentials as well. Therefore, the integral becomes \(\int \sec(u) * 2 du\).
03

Solve the integral

The integral of secant function is already known which is \[\ln|\sec u+\tan u|\]. Therefore, the solution for \(\int \sec(u) * 2 du\) is \(2\ln|\sec u+\tan u|\)+C.
04

Substitute back the original variable

Now replace \(u\) back with \(\frac{x}{2}\) to get the final solution: \(2\ln|\sec(\frac{x}{2})+\tan(\frac{x}{2})|\)+ C.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integration Techniques
Integrating functions can often seem daunting, but certain techniques simplify the process. When faced with an integral, it鈥檚 essential to identify helpful patterns or substitutions that can make the procedure easier. In our original problem, we're given the integral \[ \int \sec \frac{x}{2} \, dx \].

To tackle such integrals, understanding common functions, like trigonometric ones, and their antiderivatives is important. Methods such as substitution change variables to simplify the integration process, which can convert complex expressions into more familiar forms.
  • Substitution methods replace variables to simplify the integration.
  • Recognizing derivatives inside integrals helps to apply the Fundamental Theorem of Calculus.
  • Converting trigonometric expressions can help in integrating them more easily.
With a robust grasp of these techniques, you can approach any integral with confidence.
U-Substitution
U-substitution is a method that aids in simplifying integrals by changing variables. The key idea is to pick a substitution that will reduce the integral to a simpler form. In our original exercise, we let \( u = \frac{x}{2} \), which transforms the integral into a more manageable problem.

By substitution, \( x = 2u \) and \( dx = 2 \, du \). This conversion makes the integral easier by changing it to \( \int \sec(u) \times 2 \, du \). This not only simplifies the function but also adjusts for any constant factors as seen with the \( dx = 2 \, du \).
  • Select a substitution to simplify expressions.
  • Transform all parts of the integral, including differential \( dx \).
  • Always revert back to the original variable once the integration is completed.
Mastering u-substitution paves the way for tackling even more complex integrals.
Trigonometric Integrals
Trigonometric integrals involve functions like sine, cosine, and secant. These integrals can pose a challenge but, with a few tricks, they can be managed efficiently. The cornerstone for solving them is knowing the antiderivatives of basic trigonometric functions.

For our problem, integrating \( \sec(u) \) leads to the expression \[ \ln|\sec u + \tan u| \]. This antiderivative is critical in arriving at the solution: \( 2 \ln|\sec(\frac{x}{2}) + \tan(\frac{x}{2})| + C \).
  • Familiarize yourself with the antiderivatives of trigonometric functions.
  • Look out for identities simplifying the integral calculation.
  • Know that certain trigonometric integrals have known forms to help you solve more quickly.
Understanding these fundamentals allows us to turn complex trigonometric integrals into simpler, solvable problems.

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Most popular questions from this chapter

In Exercises 55 and 56, a differential equation, a point, and a slope field are given. (a) Sketch two approximate solutions of the differential equation on the slope field, one of which passes through the given point. (To print an enlarged copy of the graph, go to MathGraphs.com.) (b) Use integration and the given point to find the particular solution of the differential equation and use a graphing utility to graph the solution. Compare the result with the sketch in part (a) that passes through the given $$\frac{d y}{d x}=\frac{2}{\sqrt{25-x^{2}}}, \quad(5, \pi)$$

Modeling Data A valve on a storage tank is opened for 4 hours to release a chemical in a manufacturing process. The flow rate \(R\) (in liters per hour) at time \(t\) (in hours) is given in the table. \(\begin{array}{|c|c|c|c|c|c|}\hline t & {0} & {1} & {2} & {3} & {4} \\ \hline R & {425} & {240} & {118} & {71} & {36} \\ \hline\end{array}\) (a) Use the regression capabilities of a graphing utility to find a linear model for the points \((t, \ln R) .\) Write the resulting equation of the form \(\ln R=a t+b\) in exponential form. (b) Use a graphing utility to plot the data and graph the exponential model. (c) Use a definite integral to approximate the number of liters of chemical released during the 4 hours.

Evaluating a Definite Integral In Exercises \(77-80\) , evaluate the definite integral. Use a graphing utility to verify your result. $$\int_{-4}^{4} 3^{x / 4} d x$$

In Exercises 43-46, use the specified substitution to find or evaluate the integral. $$\begin{array}{l}{\int_{0}^{1} \frac{d x}{2 \sqrt{3-x} \sqrt{x+1}}} \\\ {u=\sqrt{x+1}}\end{array}$$

Finding an Indefinite Integral In Exercises \(69-76,\) find the indefinite integral. $$\int(4-x) 6^{(4-x)^{2}} d x$$
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