Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 7: Problem 24
Perform the indicated integrations. \(\int \frac{x^{3}}{x^{4}+4} d x\)
Short Answer
Expert verified
The integral is \(\frac{1}{4} \ln |x^4 + 4| + C\).
Step by step solution
01
Identify substitution
To make the integration easier, look for a substitution that simplifies the denominator. Notice that the derivative of the denominator, \(x^4 + 4\), is related to the numerator. Let's set \(u = x^4 + 4\).
02
Calculate derivative of substitution
Differentiate \(u\) with respect to \(x\): \(\frac{du}{dx} = 4x^3\). This implies that \(du = 4x^3 dx\).
03
Solve for dx
From \(du = 4x^3 dx\), solve for \(dx\). This gives \(dx = \frac{du}{4x^3}\).
04
Substitute in the integral
Substitute \(u = x^4 + 4\) and \(dx = \frac{du}{4x^3}\) into the integral. The integral becomes \(\int \frac{x^3}{u} \cdot \frac{du}{4x^3} = \frac{1}{4} \int \frac{1}{u} du\).
05
Integrate with respect to u
The integral \(\frac{1}{4} \int \frac{1}{u} du\) is a standard natural logarithm integral. It is equal to \(\frac{1}{4} \ln |u| + C\), where \(C\) is the constant of integration.
06
Back-substitute for u
Replace \(u\) with \(x^4 + 4\) in the solution, which gives \(\frac{1}{4} \ln |x^4 + 4| + C\).
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 Integrals
Definite integrals are used to calculate the accumulated sum of a function within a specific interval. They have clear lower and upper limits, which define the range of integration. When you compute a definite integral, you'll be determining the net area between the curve of the function and the x-axis over the given interval. It is represented as \ \( \int_{a}^{b} f(x) \, dx \) where \(a\) and \(b\) are the limits of integration.
The outcome of a definite integral is a specific number that describes this accumulated quantity. Some key points to remember about definite integrals include:
The outcome of a definite integral is a specific number that describes this accumulated quantity. Some key points to remember about definite integrals include:
- The Fundamental Theorem of Calculus connects differentiation and integration, asserting that integration is essentially the inverse operation of differentiation.
- They are practical for solving real-world problems, such as finding distance or area.
- Unlike indefinite integrals, they do not include a constant of integration because they yield a numerical result.
Indefinite Integrals
Indefinite integrals represent a family of functions and are used to find antiderivatives. They don鈥檛 have specific bounds like definite integrals and thus include a constant of integration, denoted as \(C\), because the antiderivative is not unique. The general form for an indefinite integral is \ \( \int f(x) \, dx = F(x) + C \), where \(F(x)\) is any antiderivative of \(f(x)\).
In the original exercise, the indefinite integral \( \int \frac{x^{3}}{x^{4}+4} dx \) was evaluated by substitution, resulting in \( \frac{1}{4} \ln |x^4 + 4| + C \). Some critical aspects of indefinite integrals include:
In the original exercise, the indefinite integral \( \int \frac{x^{3}}{x^{4}+4} dx \) was evaluated by substitution, resulting in \( \frac{1}{4} \ln |x^4 + 4| + C \). Some critical aspects of indefinite integrals include:
- They signify the reverse process of differentiation.
- The constant \(C\) represents the infinite number of antiderivatives available for a function.
- They help solve differential equations and model natural phenomena.
Calculus Techniques
Calculus techniques, such as substitution, are invaluable for solving complex integrals. Substitution is a method used to simplify integrals, making them more manageable to solve. This technique works by changing variables, essentially transforming a challenging integral into a simpler, more familiar form. The process typically involves these steps:
- Select an appropriate substitution \(u = g(x)\) that simplifies the integral, often involving setting the substitution to part of the integrand that resembles a derivative of something present in the function.
- Differentiate \(u\) with respect to \(x\) to find \(du\), and solve for \(dx\).
- Rewrite the integral in terms of \(u\), thus reducing the complexity of the problem.
