Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 28: Problem 12
Integrate each of the given functions. $$\int_{0}^{\pi / 4} \frac{2 \sec ^{2} x d x}{4+\tan x}$$
Short Answer
Expert verified
The integral evaluates to \( \ln \frac{25}{16} \).
Step by step solution
01
Recognize the Integration Technique
The given integral is \(\int_{0}^{\pi / 4} \frac{2 \sec ^{2} x \, dx}{4+\tan x}\). Notice that the derivative of \(\tan x\) is \(\sec^2 x\). This suggests that a substitution method might be effective by using \( u = \tan x \).
02
Perform the Substitution
Let \( u = \tan x \). Then, the derivative of \( u \) with respect to \( x \) is \( du = \sec^2 x \, dx \). Now, substitute \( u \) into the integral: \[\int \frac{2 \sec^2 x \, dx}{4 + \tan x} = \int \frac{2 \, du}{4 + u}\].Also, change the limits of integration. When \( x = 0 \), \( u = \tan(0) = 0 \). When \( x = \frac{\pi}{4} \), \( u = \tan\left(\frac{\pi}{4}\right) = 1 \). So the limits change from 0 to 1.
03
Integrate the Transformed Function
The integral now transforms to:\[ \int_{0}^{1} \frac{2 \, du}{4 + u} \].This is a standard form and integrates to:\[ 2 \ln|4 + u| \bigg|_{0}^{1} \].
04
Evaluate the Definite Integral
Substituting the limits of integration:\[ 2 \ln|4 + 1| - 2 \ln|4 + 0| = 2 \ln 5 - 2 \ln 4 \].Using the property of logarithms that \( a \ln b = \ln b^a \) and \( \ln a - \ln b = \ln \frac{a}{b} \), we get:\[ \ln \left(\frac{5^2}{4^2}\right) = \ln \left(\frac{25}{16}\right) \].
05
Simplify the Result
Thus, the value of the definite integral is \( \ln \frac{25}{16} \).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Integration Techniques
Integration is a central topic in calculus that involves finding the antiderivative of a function. There are several techniques available to carry out integration, making it adaptable for different types of functions.
In this exercise, the function is a rational expression containing trigonometric components such as \( \sec^2 x \) and \( \tan x \).
Some common techniques include:
In this exercise, the function is a rational expression containing trigonometric components such as \( \sec^2 x \) and \( \tan x \).
Some common techniques include:
- Substitution, where a substitution simplifies the integrand.
- Integration by parts, useful when the integrand is a product of two functions.
- Partial fraction decomposition, which breaks down the function into a sum of simpler fractions.
Substitution Method
The substitution method is a popular technique to simplify the integration process. It involves substituting part of the integrand with a new variable, often referred to as \( u \).
This step reduces the complexity by transforming the integral into a form that is easier to evaluate.
Let's explore how the substitution method was applied in this problem:
This step reduces the complexity by transforming the integral into a form that is easier to evaluate.
Let's explore how the substitution method was applied in this problem:
- We set \( u = \tan x \) because the derivative of \( \tan x \) is \( \sec^2 x \), which appears in the integrand.
- As \( du = \sec^2 x \, dx \), the integral transforms accordingly to \[ \int \frac{2}{4 + u} \, du \].
- The limits of the integration also change. When \( x=0 \), \( u=\tan(0)=0 \), and when \( x=\frac{\pi}{4} \), \( u=\tan(\frac{\pi}{4})=1 \).
Definite Integral
In calculus, a definite integral evaluates the area under a curve between two specific limits. It yields a numerical value rather than an expression.
For the transformed integral \[ \int_{0}^{1} \frac{2 \, du}{4 + u} \], we apply the standard properties of integration to achieve a result.
The steps involved are:
For the transformed integral \[ \int_{0}^{1} \frac{2 \, du}{4 + u} \], we apply the standard properties of integration to achieve a result.
The steps involved are:
- Integrate the function to get \( 2 \ln|4 + u| \big|_{0}^{1} \).
- Substitute the values of the limits to find the final result.
Calculus Education
Understanding and mastering calculus concepts like integration is vital for mathematical fluency. It forms a foundation upon which many real-world applications are built.
In calculus education, applying techniques such as substitution not only deepens comprehension but also enhances problem-solving skills.
Important aspects of learning calculus include:
In calculus education, applying techniques such as substitution not only deepens comprehension but also enhances problem-solving skills.
Important aspects of learning calculus include:
- Developing the ability to recognize which integration techniques to apply in various scenarios.
- Fostering critical thinking to manipulate and simplify complex mathematical expressions.
- Practical application of calculus in areas such as physics, engineering, and economics.
