Chapter 1: Problem 37
In the following exercises, evaluate the definite integral. $$ \int_{0}^{\pi / 4} \tan x d x $$
Short Answer
Expert verified
\( \ln \sqrt{2} \)
Step by step solution
01
Identify the integral
We need to evaluate the definite integral \( \int_{0}^{\pi / 4} \tan x \, dx \). This means we integrate the function \( \tan x \) with respect to \( x \) from \( 0 \) to \( \frac{\pi}{4} \).
02
Rewrite the tangent function
Recall that \( \tan x = \frac{\sin x}{\cos x} \). By rewriting \( \tan x \) in terms of sine and cosine, we recognize that the integral is \( \int \frac{\sin x}{\cos x} dx \).
03
Use substitution method
Let \( u = \cos x \). Then, \( \frac{du}{dx} = -\sin x \), thus \( du = -\sin x \, dx \) or \( -du = \sin x \, dx \). Substitute \( u \) into the integral to get \( \int \frac{\sin x}{\cos x} dx = \int \frac{-du}{u} \).
04
Integrate using natural logarithm
The integral \( \int \frac{-du}{u} \) is a standard integral and equals \( -\ln |u| + C \). Substitute back \( u = \cos x \), resulting in \( -\ln |\cos x| + C \).
05
Evaluate the definite integral
We evaluate \( \left. -\ln |\cos x| \right|_{0}^{\pi / 4} \). First calculate at the upper limit: \( -\ln |\cos(\pi/4)| = -\ln \left(\frac{1}{\sqrt{2}}\right) = \ln \sqrt{2} \) because \( -\ln \frac{1}{\sqrt{2}} = \ln 2^{1/2} = \ln \sqrt{2} \).
06
Simplify the evaluated expression
Next, calculate at the lower limit: \( -\ln |\cos(0)| = -\ln 1 = 0 \). The definite integral is then \( \ln \sqrt{2} - 0 = \ln \sqrt{2} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Trigonometric Integrals
In calculus, trigonometric integrals involve the integration of trigonometric functions like sine, cosine, tangent, and others. These integrals appear frequently in calculus and are often tricky due to the oscillating nature of these functions. Understanding how to rewrite or decompose a trigonometric function can simplify the integration process significantly.
For example, when working with the integral of \( \tan(x) \), a common strategy is to rewrite it using other trigonometric functions, like sine and cosine. Recall that tangent, \( \tan(x)=\frac{\sin(x)}{\cos(x)} \), can be decomposed into sine and cosine. This decomposition changes the function into a form more suitable for different integration techniques.
Rewriting and breaking down trigonometric functions is essential because some forms allow for easier manipulation or substitution, which can turn a challenging problem into a manageable one. Remembering these identities and being able to use them flexibly is crucial when tackling trigonometric integrals.
For example, when working with the integral of \( \tan(x) \), a common strategy is to rewrite it using other trigonometric functions, like sine and cosine. Recall that tangent, \( \tan(x)=\frac{\sin(x)}{\cos(x)} \), can be decomposed into sine and cosine. This decomposition changes the function into a form more suitable for different integration techniques.
Rewriting and breaking down trigonometric functions is essential because some forms allow for easier manipulation or substitution, which can turn a challenging problem into a manageable one. Remembering these identities and being able to use them flexibly is crucial when tackling trigonometric integrals.
Substitution Method
The substitution method, also known as \( u \)-substitution, is a powerful tool for simplifying integrals, especially when involving products or quotients of functions. This technique involves changing the variable of integration to transform the integral into a simpler form.
In the given exercise, to integrate \( \int \frac{\sin x}{\cos x}dx \), we use substitution. We set \( u = \cos x \), which implies \( du = -\sin x \, dx \). This substitution method is particularly useful, as it changes the original complex integral into a more familiar form: \( \int \frac{-du}{u} \).
This step then allows us to integrate using a simple natural logarithm function, making the once daunting integral manageable. It's crucial to remember to change the limits of integration accordingly if they are definite, though in this specific exercise, focusing on the indefinite form initially is helpful before returning to the original limits of \( x \).
The core idea is that substitution can transform an unfamiliar or complex integral into something straightforward that we can solve with 'basic' integration rules.
In the given exercise, to integrate \( \int \frac{\sin x}{\cos x}dx \), we use substitution. We set \( u = \cos x \), which implies \( du = -\sin x \, dx \). This substitution method is particularly useful, as it changes the original complex integral into a more familiar form: \( \int \frac{-du}{u} \).
This step then allows us to integrate using a simple natural logarithm function, making the once daunting integral manageable. It's crucial to remember to change the limits of integration accordingly if they are definite, though in this specific exercise, focusing on the indefinite form initially is helpful before returning to the original limits of \( x \).
The core idea is that substitution can transform an unfamiliar or complex integral into something straightforward that we can solve with 'basic' integration rules.
Integration Techniques
Integration techniques are the methods and strategies used to find integrals, essential for solving calculus problems. Several standard techniques include integration by parts, substitution, partial fractions, and trigonometric identities.
In the context of solving \( \int_{0}^{\pi/4} \tan x \, dx \), we utilized trigonometric identities and the substitution method. First, we rewrote \( \tan(x) \) in terms of sine and cosine, then applied substitution to simplify the integration process. Such techniques are crucial as they provide systematic ways to approach and solve various integrals.
In this specific exercise, once substitution was applied, integrating \( \frac{-du}{u} \) was straightforward. It resulted in a natural logarithm form, specifically, \( -\ln|u| + C \).
Finally, the definite integral evaluation included calculating this integral expression at the upper and lower limits, using the properties of logarithms to arrive at a concise result. It's these techniques, along with practice, that demystify the integration process, making even complicated integrals within reach.
In the context of solving \( \int_{0}^{\pi/4} \tan x \, dx \), we utilized trigonometric identities and the substitution method. First, we rewrote \( \tan(x) \) in terms of sine and cosine, then applied substitution to simplify the integration process. Such techniques are crucial as they provide systematic ways to approach and solve various integrals.
In this specific exercise, once substitution was applied, integrating \( \frac{-du}{u} \) was straightforward. It resulted in a natural logarithm form, specifically, \( -\ln|u| + C \).
Finally, the definite integral evaluation included calculating this integral expression at the upper and lower limits, using the properties of logarithms to arrive at a concise result. It's these techniques, along with practice, that demystify the integration process, making even complicated integrals within reach.
