/*! 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 27 If \(\mathrm{f}(\mathrm{x})\) is... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 27

If \(\mathrm{f}(\mathrm{x})\) is a non-negative continuous function such that \(\mathrm{f}(\mathrm{x})+\mathrm{f}(\mathrm{x}+1 / 2)=1\), then find the value of \(\int_{0}^{2} f(x) d x\)

Short Answer

Expert verified
Answer: The value of the integral 鈭個虏 f(x) dx is 1.

Step by step solution

01

Express f(x) using the given condition

Since we have the condition \(f(x) + f(x + \frac{1}{2}) = 1\), we can express f(x) as \(f(x) = 1 - f(x + \frac{1}{2})\).
02

Substitute f(x) in the integral

Now we need to find the value of \(\int_{0}^{2} f(x) dx\). Using the expression we found in step 1, the integral becomes: \(\int_{0}^{2} (1 - f(x + \frac{1}{2})) dx\).
03

Use substitution method for integration

Let's make the substitution \(u = x + \frac{1}{2}\). Then, \(du = dx\) and when \(x = 0\), \(u = \frac{1}{2}\) and when \(x = 2\), \(u = \frac{5}{2}\). Therefore, the integral becomes: \(\int_{\frac{1}{2}}^{\frac{5}{2}} (1 - f(u)) du\).
04

Split the integral

Now, we split the integral into two parts and solve them separately: $$\int_{\frac{1}{2}}^{\frac{5}{2}} (1 - f(u)) du = \int_{\frac{1}{2}}^{\frac{5}{2}} 1 du - \int_{\frac{1}{2}}^{\frac{5}{2}} f(u) du.$$
05

Evaluate the first integral

The first integral is quite simple: $$\int_{\frac{1}{2}}^{\frac{5}{2}} 1 du = u \Big|_{\frac{1}{2}}^{\frac{5}{2}} = \frac{5}{2} - \frac{1}{2} = 2.$$
06

Recognize that the second integral is same as the original integral

Observe that the second integral is equivalent to the original integral, but with different limits of integration: $$\int_{\frac{1}{2}}^{\frac{5}{2}} f(u) du = \int_{0}^{2} f(x) dx.$$ Therefore, we can write the entire equation as: $$\int_{0}^{2} f(x) dx = 2 - \int_{0}^{2} f(x) dx.$$
07

Solve for the original integral

To find the value of \(\int_{0}^{2} f(x) dx\), we can now add the integral to both sides of the equation: $$2 \int_{0}^{2} f(x) dx = 2.$$ Finally, divide both sides by 2: $$\int_{0}^{2} f(x) dx = 1.$$ The value of the integral \(\int_{0}^{2} f(x) dx\) is 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!

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

4\. Prove that (i) \(\frac{2 \pi}{13}<\int_{0}^{2 \pi} \frac{\mathrm{dx}}{10+3 \cos \mathrm{x}}<\frac{2 \pi}{7}\) (ii) \(0<\int_{0}^{\pi / 4} x \sqrt{\tan x}<\frac{\pi^{2}}{32}\) (iii) \(\frac{1}{2}<\int_{\pi / 4}^{\pi / 2} \frac{\sin \mathrm{x}}{\mathrm{x}} \mathrm{dx}<\frac{1}{\sqrt{2}}\) (iv) \(\left|\int_{1}^{4} \frac{\sin x}{x} d x\right| \leq \frac{3}{2}\).

Prove that \(\int_{0}^{1} x^{n} \ln x d x=\frac{1}{(n+1)^{2}}, \quad n>-1\)

\begin{aligned} &\text { Integrating by parts, prove that }\\\ &0<\int_{100 \pi}^{200 \pi} \frac{\cos \mathrm{x}}{\mathrm{x}} \mathrm{dx}<\frac{1}{100 \pi} \end{aligned}

For each \(x>0 .\) let \(G(x)\) \(=\int_{0}^{\infty} \mathrm{e}^{-\mathrm{xt}} \mathrm{dt}\). Prove that \(\mathrm{xG}(\mathrm{x})=1\) for each \(\mathrm{x}>0\).

Prove that \(\int_{a}^{b} \frac{d x}{\sqrt{\\{(x-a)(b-x)\\}}}=\pi\), \(\int_{a}^{b} \frac{x d x}{\sqrt{\\{(x-a)(b-x)\\}}}=\frac{1}{2} \pi(a+b)\) (i) by means of the substitution \(\mathrm{x}=\mathrm{a}+(\mathrm{b}-\mathrm{a}) \mathrm{t}^{2}\), (ii) bymeans of the substitution \((\mathrm{b}-\mathrm{x})(\mathrm{x}-\mathrm{a})=\mathrm{t}\), and (iii) by means of the substitution \(x=a \cos ^{2} t\) \(+b \sin ^{2} t\)
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

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.