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Chapter 1: Problem 297

In the following exercises, use a change of variables to evaluate the definite integral. $$ \int_{0}^{\pi / 4} \frac{\sin \theta}{\cos ^{4} \theta} d \theta $$

Short Answer

Expert verified
The definite integral evaluates to \( \frac{8\sqrt{2} - 8}{3} \).

Step by step solution

01

Identify the substitution

To make the integral easier to solve, let's use a substitution. We will set \( u = \cos \theta \). Therefore, the derivative with respect to \( \theta \) is \( du = -\sin \theta \, d\theta \). Thus, \( \sin \theta \, d\theta = -du \).
02

Change limits of integration

Since the integral is a definite integral, we need to change the limits of integration according to our substitution. When \( \theta = 0 \), \( u = \cos(0) = 1 \). When \( \theta = \frac{\pi}{4} \), \( u = \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \). Thus, the new limits for \( u \) are from 1 to \( \frac{\sqrt{2}}{2} \).
03

Substitute and simplify the integral

Substitute \( u \) and \( du \) into the integral: \[\int_{1}^{\frac{\sqrt{2}}{2}} \frac{-1}{u^{4}} \, du.\] We can rewrite this as \[-\int_{1}^{\frac{\sqrt{2}}{2}} u^{-4} \, du.\]
04

Evaluate the integral

Now, integrate \( -\int u^{-4} \, du \). The integral of \( u^{-4} \) is \( \frac{u^{-3}}{-3} = -\frac{1}{3} u^{-3} \). Thus, we have:\[-\left[ -\frac{1}{3} u^{-3} \right]_{1}^{\frac{\sqrt{2}}{2}} = \left[ \frac{1}{3} u^{-3} \right]_{1}^{\frac{\sqrt{2}}{2}}\]
05

Substitute back the limits

Substitute the limits back into the antiderivative:\[\frac{1}{3} \left( \left(\frac{\sqrt{2}}{2}\right)^{-3} - 1^{-3} \right).\] Simplify this to get:\[\frac{1}{3} \left( \left(\frac{2}{\sqrt{2}}\right)^{3} - 1 \right).\] This simplifies further to evaluate to:\[\frac{1}{3} \left( 2\sqrt{2}-1 \right).\]
06

Final Step: Simplify the result

Finally, simplify to find\[\frac{1}{3} (16\sqrt{2} - 8) = \frac{8\sqrt{2} - 8}{3} \approx 1.77124.\]

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Key Concepts

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

Change of Variables
When faced with a complex definite integral, the technique of change of variables can be particularly useful. This entails substituting a part of the integral with a new variable. In our example, the substitution chosen was \( u = \cos \theta \). This substitution simplifies the original integral by offering another perspective on both the function and its bounds, often turning a complicated expression into a more manageable one. Steps to apply a substitution:
  • Identify a suitable part of the integral to substitute, which simplifies the mathematical expression.
  • Find the derivative of the substitution with respect to the original variable.
  • Use the relationship from the derivative to replace the corresponding differential element in the expression.
A well-chosen substitution turns an intimidating problem into an approachable one, paving the way for further manipulation and evaluation.
Trigonometric Substitution
Trigonometric substitution is a valuable method when dealing with integrals involving trigonometric functions. This is especially true for expressions where trigonometric identities can transform the function into a simpler computable form. In the given problem, recognizing \( \sin \theta \) and \( \cos \theta \) allowed us to implement a substitution with \( u = \cos \theta \). Unified approach:
  • This substitution transforms the sine function into \(-du\), simplifying the integral's form.
  • By relating trigonometric functions via identities, like \( \sin^2 \theta + \cos^2 \theta = 1 \), we highlight the interdependencies of the functions.
  • This makes new lower and upper bounds derivable and sets the stage for the next steps in integration.
For those struggling with trigonometric integrals, understanding these relationships and substitutions provides a clearer, more concise path to the solution.
Limits of Integration
Once a substitution is made in a definite integral, the original limits of integration change. This aspect is crucial; otherwise, the integration results will be inaccurate. For our exercise, the substitution from \( \theta \) to \( u \) affected both the lower and upper limits. Adjusting the limits involves:
  • Evaluating the new variable \( u \) at the original limits: when \( \theta = 0 \), \( u = \cos(0) = 1 \), and when \( \theta = \frac{\pi}{4} \), \( u = \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \).
  • Replacing the original integral limits with these newly calculated values for \( u \).
This change allows the integral to be evaluated precisely over the correct interval, maintaining the integrity and accuracy of the integration process.
Integration Techniques
Successfully evaluating a definite integral requires the use of various integration techniques that simplify the process. In our problem, we employed substitution, integration rules, and trigonometric identities. Key techniques involve:
  • Substituting to transform the integral into a more manageable format, such as from \( \theta \) to \( u \).
  • Utilizing fundamental integration rules to compute standard forms, e.g., integrating \( u^{-4} \) to get \(-\frac{1}{3} u^{-3} \).
  • Simplifying expressions through algebraic manipulation, finally substituting back the limits to arrive at the precise value of the integral.
Combining these methods efficiently leads to a more straightforward and coherent evaluation of the integral, emphasizing the importance of understanding and selecting the right technique for each step.

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