/*! 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 46 Evaluate the definite integral. ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 46

Evaluate the definite integral. $$ \int_{2}^{4} \frac{x^{2}}{(3 x-5)} d x $$

Short Answer

Expert verified
The exact value of the integral is \( 44 \).

Step by step solution

01

Set the Value of U

Set the substitution variable, \( u = 3x - 5 \), which simplifies the denominator.
02

Find the value of dx

Differentiate \( u = 3x - 5 \) with respect to \( x \) to find \( du/dx = 3 \), from which we get \( dx = du/3 \).
03

Substitute in the Integral

Substitute \( u \) and \( dx \) in the integral to get a new integral: \( \int_{1}^{3} \frac{(u + 5)^2}{u} * (du/3) \).
04

Simplify the Integral

Simplify the integrand to get \( \int_{1}^{3} (u/3 + 5)^2 du \).
05

Evaluate the Integral

Evaluate the integral to obtain \( \frac{1}{9}u^3 + 5u^2 + 25u |_{1}^{3} \).
06

Substitute U Back

Substitute \( u = 3x - 5 \) back in to get \( \frac{1}{9}(3x - 5)^3 + 5x^2 + 25x \) and evaluate this from 2 to 4.
07

Final Answer

Calculate the final answer by using the fundamental theorem of calculus: \( F(4) - F(2) \) where \( F(x) = \frac{1}{9}(3x - 5)^3 + 5x^2 + 25x \).

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

Find the indefinite integral (a) using the integration table and (b) using the specified method. Integral \mathrm{Method } $$ \begin{aligned} &\int \frac{1}{x^{2}-75} d x\\\ &\text { Partial fractions } \end{aligned} $$

Present Value A business is expected to yield a continuous flow of profit at the rate of \(\$ 500,000\) per year. If money will earn interest at the nominal rate of \(9 \%\) per year compounded continuously, what is the present value of the business (a) for 20 years and (b) forever?

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. $$ \int_{0}^{2} \frac{1}{\sqrt[3]{x-1}} d x $$

Prove that Simpson's Rule is exact when used to approximate the integral of a cubic polynomial function, and demonstrate the result for \(\int_{0}^{1} x^{3} d x, n=2\).

Explain why the integral is improper and determine whether it diverges or converges. Evaluate the integral if it converses. $$ \int_{0}^{\infty} e^{-x} d x $$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Decision Maths

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.