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Chapter 7: Problem 61

Evaluate the integral. \( \displaystyle \int \frac{d \theta}{1 + \cos \theta} \)

Short Answer

Expert verified
\(\tan\left(\frac{\theta}{2}\right) + C\)

Step by step solution

01

Simplify the Denominator

Recall the trigonometric identity: \(1 + \cos \theta = 2\cos^2(\frac{\theta}{2})\). Using this identity, rewrite the denominator of the integral: \[\int \frac{d\theta}{1 + \cos \theta} = \int \frac{d\theta}{2\cos^2(\frac{\theta}{2})}\].
02

Further Simplify and Use Substitution

Simplify the expression by factoring out the \(\frac{1}{2}\) constant:\[\int \frac{d\theta}{1 + \cos \theta} = \frac{1}{2} \int \sec^2(\frac{\theta}{2}) \, d\theta\].Now, use the substitution: Let \(u = \frac{\theta}{2}\), which implies \(d\theta = 2 \, du\). Substitute these into the integral:\[\frac{1}{2} \int \sec^2(\frac{\theta}{2}) \, d\theta = \int \sec^2(u) \, du\].
03

Integrate Using Known Result

We know the integral of \(\sec^2(u)\) is \(\tan(u)\), so we get:\[\int \sec^2(u) \, du = \tan(u) + C\].
04

Substitute Back to Original Variable

Substitute back the original variable: Recall that \(u = \frac{\theta}{2}\), so:\[\tan(u) = \tan\left(\frac{\theta}{2}\right)\].Therefore, the final answer in terms of \(\theta\) is:\[\int \frac{d\theta}{1 + \cos \theta} = \tan\left(\frac{\theta}{2}\right) + C\].

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Key Concepts

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

Trigonometric Identities
Trigonometric identities are essential tools in calculus, especially when evaluating integrals involving trigonometric functions. In this exercise, we use the identity for simplifying the expression used in the integral:
  • The identity used here is: \(1 + \cos \theta = 2\cos^2\left(\frac{\theta}{2}\right)\). This specific identity helps transform the integral into a form that's easier to work with.
  • Trigonometric identities often relate different trigonometric functions such as sine, cosine, and tangent. They can also simplify expressions by converting products and powers of trigonometric functions into sums or other functions.
  • By knowing and applying these identities, you can transform complex trigonometric forms into simpler ones. This is a crucial step in solving integrals effectively.
Understanding these identities not only aids in performing integrations but also improves problem-solving skills across other areas of mathematics.
Substitution Method
The substitution method is a technique used to evaluate integrals by simplifying functions into more manageable forms. Here's how it applies to our exercise:
  • Substitution is often used when you encounter an integral that seems difficult at a first glance. Instead of solving the integral directly in terms of \(\theta\), we introduce a new variable, \(u\), to simplify the task.
  • In this case, we use the substitution \(u = \frac{\theta}{2}\). This transforms both the function inside the integral and the differential \(d\theta\) into \(d\theta = 2 \, du\), simplifying the expression to a function of \(u\).
  • Substitution thereby turns the integral from one involving \(\theta\) to one in terms of \(u\). Once integration in terms of \(u\) is completed, it is crucial to substitute back to the original variable to find the final solution.
This method can significantly reduce the complexity of solving integrals, making it a vital tool for students learning integration techniques.
Integration Techniques
Different integration techniques are used to solve various forms of integrals. The exercise at hand utilizes these techniques effectively:
  • Firstly, the trigonometric identity simplifies the given integral, allowing it to be easily reshaped for substitution and integration.
  • The substitution method comes into play, transforming complex integrals into simpler ones. By rewriting the problem in terms of \(u\), the integral becomes manageable.
  • Lastly, we apply the basic integral \(\int \sec^2(u) \, du\), which is a well-known form that integrates to \(\tan(u) + C\). Such fundamental integrals are solutions you should have at your fingertips when dealing with calculus problems.
Applying these integration techniques in combination makes it possible to solve some of the more challenging integrals you might face. These methods showcase the process of breaking down the integral into smaller, more manageable steps.

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Most popular questions from this chapter

Use a computer algebra system to evaluate the integral. Compare the answer with the result of using tables. If the answers are not the same, show that they are equivalent. \( \displaystyle \int \frac{1}{\sqrt{1 + \sqrt[3]{x}}}\ dx \)

Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson's Rule to approximate the given integral with the specified value of \( n \). (Round your answers to six decimal places.) \( \displaystyle \int_0^4 \ln (1 + e^x)\ dx \) , \( n = 8 \)

Determine whether each integral is convergent or divergent. Evaluate those that are convergent. \( \displaystyle \int_2^\infty \frac{dv}{v^2 + 2v - 3} \)

If \( f(t) \) is continuous for \( t \ge 0 \), the Laplace transform of \( f \) is the function \( F \) defined by $$ F(s) = \int_0^\infty f(t) e^{-st}\ dt $$ and the domain of \( F \) is the set consisting of all numbers s for which the integral converges. Find the Laplace transforms of the following functions. (a) \( f(t) = 1 \) (b) \( f(t) = e^t \) (c) \( f(t) = t \)

The figure shows a pendulum with length \( L \) that makes a maximum angle \( \theta_0 \) with the vertical. Using Newton's Second Law, it can be shown that the period \( T \) (the time for one complete swing) is given by $$ T = 4 \sqrt{\frac{L}{g}} \int_0^{\frac{\pi}{2}} \frac{dx}{\sqrt{1 - k^2 \sin^2 x}} $$ where \( k = \sin \left (\frac{1}{2} \theta_0 \right) \) and \( g \) is the acceleration due to gravity. If \( L = 1 m \) and \( \theta_0 = 42^\circ \), use Simpson's Rule with \( n = 10 \) to find the period.
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