/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 45 Find the integral. \(\int \fra... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 45

Find the integral. \(\int \frac{x}{x^{4}+1} d x\)

Short Answer

Expert verified
The integral \(\int \frac{x}{x^{4}+1}d x = \frac{1}{4}\ln |x^{4}+1|\)

Step by step solution

01

Identify the Substitution

Let \(u=x^{4}+1\). Then, the differential of \(u\) is \(du=4x^{3}dx\). Now, it's clear from the given integral that multiplying and dividing it by 4 doesn't change its value, so we transform the original integral \(\int \frac{x}{x^{4}+1}d x\) to \(\frac{1}{4} \int \frac{4x}{x^{4}+1}d x = \frac{1}{4} \int \frac{du}{u}\).
02

Perform Integration

The integral of \(\frac{1}{u}\) with respect to \(u\) is \(\ln |u|\), therefore applying the integral gives us \(\frac{1}{4} \int \frac{du}{u} = \frac{1}{4}\ln |u|\) .
03

Substitute u back into the Integral

In step 1, we defined \(u=x^{4}+1\). We replace this back into our integral expression from step 2. Thus, our final answer will be \(\frac{1}{4}\ln |x^{4}+1|\).

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
When we encounter an integral, it often requires a bit more than straightforward application of rules. Various integration techniques exist to simplify and solve intricate integrals. These methods are tools in our calculus toolkit, such as:
  • Substitution: Useful when the integral contains a function and its derivative.
  • Integration by Parts: Ideal for products of functions, reminiscent of the product rule in differentiation.
  • Partial Fraction Decomposition: Valuable for integrals involving rational functions.
By utilizing these techniques, we can transform complicated integrals into a more manageable form. Integration often requires creativity in deciding which technique to apply, showcasing the versatility and importance of these mathematical tools. In our example with \(\int \frac{x}{x^4+1} dx\), we opted for the substitution method for simplicity.
Substitution Method
The substitution method is akin to reversing the chain rule from differentiation. It involves substituting part of the integral with a single variable, typically denoted as \(u\), to make the integral easier to solve. This method becomes especially handy when you spot a function and its derivative present within the integrand.

Here's how it works:
  • Select a substitution: Identify a part of the integrand to substitute, like \(u = x^4 + 1\) in our example.
  • Differentiate \(u\): Compute \(du\) by differentiating your substitution, in this case, \(du = 4x^3 dx\).
  • Adjust the integral: Rewrite the integral in terms of \(u\), often requiring adjustments like multiplying by constants to match the differential.
    In \(\int \frac{x}{x^4 + 1} dx\), this step results in \(\frac{1}{4} \int \frac{du}{u}\).
  • Integrate: Solve the simpler integral with respect to \(u\).
  • Back-substitute: Replace \(u\) with the original variable expressions to complete the solution.
The substitution method can greatly simplify solving integrals, making it a favorite among mathematicians and students alike.
Definite and Indefinite Integrals
In calculus, integrals come in two flavors: definite and indefinite. Each serves a different purpose but is related in concept.

**Indefinite Integrals:**
  • An indefinite integral is the most basic form and is expressed as \(\int f(x) \, dx\).
  • It represents a family of functions, or antiderivatives, that we denote with a constant of integration \(C\).
  • For instance, the solution \(\frac{1}{4}\ln|x^4 + 1| + C\) is an indefinite integral.
**Definite Integrals:**
  • They are evaluated over a specific interval \([a, b]\) and give a numerical value.
  • Mathematically represented as \(\int_{a}^{b} f(x) \, dx\), indicating the area under the curve from \(a\) to \(b\).
  • The result of a definite integral is a number, unlike the function result of an indefinite integral.
Understanding the distinction between these two forms is crucial, as it determines whether we're looking for a function or an area/value. Each has its place in broader calculus applications, from physics calculations to probability assessments.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find \(F^{\prime}(x)\). $$ F(x)=\int_{0}^{x^{3}} \sin t^{2} d t $$

Find the derivative of the function. \(y=\tanh ^{-1}(\sin 2 x)\)

Find the integral. \(\int \frac{2}{x \sqrt{1+4 x^{2}}} d x\)

Consider the integral \(\int \frac{1}{\sqrt{6 x-x^{2}}} d x\). (a) Find the integral by completing the square of the radicand. (b) Find the integral by making the substitution \(u=\sqrt{x}\). (c) The antiderivatives in parts (a) and (b) appear to be significantly different. Use a graphing utility to graph each antiderivative in the same viewing window and determine the relationship between them. Find the domain of each.

In Exercises 31 and \(32,\) show that the function satisfies the differential equation. \(y=a \sinh x\) \(y^{\prime \prime \prime}-y^{\prime}=0\)
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.