Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 28: Problem 32
Integrate each of the given functions. $$\int \frac{\sec ^{2} t \tan t}{4+\sec ^{2} t} d t$$
Short Answer
Expert verified
The integral is \( \frac{1}{2} \ln|4+\sec^2 t| + C \).
Step by step solution
01
Simplify the Integral Expression
First, observe that the integrand can be rearranged with a simple substitution in mind. We know that the derivative of \( \sec^2 t \) involves \( \sec^2 t \tan t \). Thus, we can look to perform a substitution by letting \( u = \sec^2 t \). Then, \( du = 2 \sec^2 t \tan t \, dt \). Thus, \( \sec^2 t \tan t \, dt = \frac{1}{2} du \).
02
Substitute in the Integral
Apply the substitution \( u = \sec^2 t \) to rewrite the integral: \[\int \frac{\sec^2 t \tan t}{4 + \sec^2 t} \, dt = \int \frac{1}{4+u} \cdot \frac{1}{2} du = \frac{1}{2} \int \frac{1}{4+u} \, du.\]
03
Integrate the New Function
Use the integration formula for \( \int \frac{1}{a+u} \, du = \ln|a+u| + C \), setting \( a = 4 \), to solve the integral:\[\frac{1}{2} \int \frac{1}{4+u} \, du = \frac{1}{2} \ln|4+u| + C.\]
04
Back Substitution
Return to the original variable by substituting back \( u = \sec^2 t \), giving us:\[\frac{1}{2} \ln|4+\sec^2 t| + C.\]
05
Finalize the Solution
The antiderivative of the given function is:\[\frac{1}{2} \ln|4+\sec^2 t| + C,\] where \( C \) is the constant of integration.
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.
Definite and Indefinite Integrals
In calculus, an integral represents the area under a curve, between the curve and the x-axis, over a specified interval. There are two types of integrals: definite and indefinite.
- **Definite integrals** have limits: They calculate the area under the curve between two points, providing a specific numerical value. For example, \( \int_a^b f(x) \, dx \) gives you the area from \( x=a \) to \( x=b \).
- **Indefinite integrals** lack bounds: They represent families of functions and include a constant of integration \( C \). Thus, \( \int f(x) \, dx \) provides an antiderivative of the function \( f(x) \).
Substitution Method
The substitution method is a powerful tool for solving integrals, especially when the integrand is complex. It's comparable to using the chain rule in reverse.
To employ substitution, you follow these steps:
To employ substitution, you follow these steps:
- Identify a substitution component: Look for part of the integral whose derivative is present elsewhere in the integrand. In our problem, we choose \( u = \sec^2 t \) because its derivative \( du = 2\sec^2 t \tan t \, dt \) is related to the integral.
- Rewrite the integral: Replace all instances of the selected component with \( u \) and adjust \( dt \) appropriately, simplifying the integral. Here, \( \int \frac{\sec^2 t \tan t}{4+\sec^2 t} \, dt \) becomes \( \frac{1}{2} \int \frac{1}{4+u} \, du \).
- Solve and substitute back: Find the integral in terms of \( u \), then replace \( u \) with its equivalent term back in the original variable. This ensures you return to the function鈥檚 original context.
Trigonometric Functions
Trigonometric functions, like \( \sec t \) and \( \tan t \), arise frequently in calculus problems, especially in integrals. Recognizing their derivatives and function relationships is crucial.
- The function \( \sec t \) is the reciprocal of \( \cos t \), specifically \( \sec t = \frac{1}{\cos t} \). When differentiated, it gives \( \sec t \tan t \).
- The tangent function, \( \tan t \), is the ratio of \( \sin t \) to \( \cos t \), or \( \tan t = \frac{\sin t}{\cos t} \). Its derivative is \( \sec^2 t \).
Calculus Steps
Solving calculus problems involves clear, methodical steps. This step-by-step approach ensures you don't miss crucial elements of the process. Let's revisit the essential steps for solving our integral.**Step-by-step process:**
- **Simplify**: Look for ways to simplify the integrand by identifying possible substitutions. This often involves recognizing derivatives within the integral.
- **Substitute**: Apply your identified substitution, re-expressing much of the integrand to the new variable, which often simplifies the integration drastically.
- **Integrate**: Perform the integration using known formulas or basic integration methods. In our solution, integrating \( \int \frac{1}{4+u} \, du \) resulted in \( \ln|4+u| + C \).
- **Back-substitute**: Replace your substitute variable back with the original variable to return to the original expression's context. In our example, \( u = \sec^2 t \).
- **Finalize**: Simplifying and ensuring correct recognition of the constant of integration \( C \), as seen in the final form: \( \frac{1}{2} \ln|4+\sec^2 t| + C \).
