/*! 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 8 Show that the integration by par... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 8

Show that the integration by parts formula of extends to the case where \(f\) and \(g\) are continuous and \(f^{\prime}\) and \(g^{\prime}\) are Riemann integrable.

Short Answer

Expert verified
The integration by parts formula holds when \( f \) and \( g \) are continuous and their derivatives are Riemann integrable.

Step by step solution

01

Recall the Integration by Parts Formula

The integration by parts formula is given by:\[ \int_a^b f(x)g'(x) \, dx = \left[ f(x)g(x) \right]_a^b - \int_a^b f'(x)g(x) \, dx. \]This is a powerful tool in calculus used to transform the integral of a product of functions into a potentially simpler form.
02

Conditions of Continuity and Integrability

We need to apply this formula under the assumption that the functions \( f \) and \( g \) are continuous, while their derivatives \( f' \) and \( g' \) are Riemann integrable. These conditions ensure that both original functions are well-behaved across the interval \([a, b]\) and that their derivatives can be integrated over this interval.
03

Apply Integration by Parts Formula

Even with these conditions, the integration by parts formula is applicable. Since \( f \) and \( g \) are continuous, their product \( f(x)g(x) \) will also be continuous. This continuity ensures that the term \( \left[ f(x)g(x) \right]_a^b \) can be evaluated at the bounds \( a \) and \( b \). Since \( g' \) and \( f' \) are Riemann integrable, the integrals \( \int_a^b f(x)g'(x) \ dx \) and \( \int_a^b f'(x)g(x) \ dx \) both exist.
04

Conclusion

Since the assumptions of continuity and integrability hold, the integration by parts formula remains valid in this scenario. The boundary evaluations and integrals remain well-defined and calculable.

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.

Riemann integrable functions
The concept of Riemann integrability is crucial in understanding how certain functions can be integrated over a given interval. A function is Riemann integrable if the set of its discontinuities has measure zero. This means the function can be approximated closely by sums of its values over small subintervals.
For most practical purposes, if a function is continuous or has only isolated discontinuities, it is Riemann integrable. In the context of integration by parts, having derivatives like \( f' \) and \( g' \) be Riemann integrable ensures that their integrals exist and can be calculated over a defined interval such as \([a, b]\).
This integrability condition is important because it allows us to compute integrals and apply the formula of integration by parts without encountering undefined or infinite results. It guarantees that our mathematical operations lead to meaningful outcomes.
Continuous functions
Continuous functions are a central concept in calculus and critical for applying techniques like integration by parts. A function is continuous at a point if the limit of the function as it approaches the point is equal to its value at that point. Essentially, there are no jumps, breaks, or holes in the graph of a continuous function.
For integration by parts, the continuous nature of functions \( f \) and \( g \) ensures that their product \( f(x)g(x) \) is also continuous. This is essential because the term \( \left[ f(x)g(x) \right]_a^b \) involves evaluating the product at the boundary points \( a \) and \( b \).
Without continuity, assessing this product at specific points could be problematic or even undefined if there were abrupt changes in function behavior. Continuity also aids in ensuring the overall integrity of the integral solving process.
Calculus techniques
Calculus techniques provide numerous tools for solving complex problems involving functions and their integrals or derivatives. One such technique is integration by parts, which stems from the product rule for differentiation. This technique is incredibly useful for transforming a complex integral into a more manageable form.
The fundamental formula \( \int_a^b f(x)g'(x) \ dx = \left[ f(x)g(x) \right]_a^b - \int_a^b f'(x)g(x) \ dx \) allows us to simplify the integration process, especially when dealing with products of functions.
In practice, calculus techniques like this require careful consideration of function properties such as continuity and Riemann integrability, as these affect the validity and simplicity of the operations. By mastering these techniques, one can solve a broader range of mathematical problems effectively.

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

For what values of \(p, q\) are the integrals $$ \int_{0}^{1} \frac{\sin x}{x^{p}} d x \text { and } \int_{0}^{1} \frac{(\sin x)^{q}}{x} d x $$ ordinary Riemann integrals, convergent improper Riemann integrals, or divergent improper Riemann integrals?

Let \(f\) be a continuous function on \([1, \infty)\) such that \(\lim _{x \rightarrow \infty} f(x)=\alpha\). Show that if the integral \(\int_{1}^{\infty} f(x) d x\) converges, then \(\alpha\) must be \(0 .\)

Let \(f\) be bounded on \([a, b]\) and continuous a.e. on \([a, b] .\) Suppose that \(F\) is defined on \([a, b]\) and that \(F^{\prime}(x)=f(x)\) for all \(x\) in \([a, b]\) except at the points of some set of measure zero. (a) Is it necessarily true that \(F(x)-F(a)=\int_{a}^{x} f(t) d t\) for every \(x \in[a, b] ?\) (b) Same question as in (a) but assume also that \(F\) is continuous. (c) Same question, but this time assume that \(F\) is a Lipschitz function. You may assume the nonelementary fact that a Lipschitz function \(H\) with \(H^{\prime}=0\) a.e. must be constant. (d) Give an example of a Lipschitz function \(F\) such that \(F\) is differentiable, \(F^{\prime}\) is bounded, but \(F^{\prime}\) is not integrable.

Suppose that the oscillation \(\omega_{f}(x)\) of a function \(f\) is smaller than \(\eta\) at each point \(x\) of an interval \([c, d]\). Show that there must be a partition \(\left[x_{0}, x_{1}\right]\), \(\left[x_{1}, x_{2}\right], \ldots,\left[x_{n-1}, x_{n}\right]\), of \([c, d]\) so that the oscillation $$ \omega f\left(\left[x_{k-1}, x_{k}\right]\right)<\eta $$ on each member of the partition.

Formulate a definition of the integral \(\int_{-\infty}^{b} f(x) d x\) for a function continuous on \((-\infty, b] .\) Supply examples of convergent and divergent integrals of this type.
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

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.