Answered

Welcome to Westonci.ca, where you can find answers to all your questions from a community of experienced professionals. Discover comprehensive answers to your questions from knowledgeable professionals on our user-friendly platform. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.

This question is designed to be answered without a calculator. Use the antiderivative formula shown, where f(u) represents a function. Integral of (l n u) d u = u ln u – f(u) + C Which function represents f(u)?

Sagot :

MrRoyal

Answer:

[tex]f(u) = u[/tex]

Step-by-step explanation:

Given

[tex]\int {\ln(u)} \, du = u[\ln(u) - f(u)] + c[/tex]

Required

Find f(u)

To do this, we start by integrating the left-hand side

[tex]\int {\ln(u)} \, du = u\ln(u) - f(u)+ c[/tex]

Using integration by parts, we have:

[tex]\int {fg'} = fg - \int f'g[/tex]

So, we have:

[tex]f = \ln(u)[/tex]

Differentiate

[tex]f'=\frac{1}{u}[/tex]

[tex]g' = 1[/tex]

Integrate

[tex]g=u[/tex]

So:

[tex]\int {fg'} = fg - \int f'g[/tex]

[tex]\int \ln(u)\ du = \ln(u) * u - \int \frac{1}{u} * u\ du[/tex]

[tex]\int \ln(u)\ du = u\ln(u) - \int \ du[/tex]

So, we have:

[tex]u\ln(u) - \int \ du = u\ln(u) - f(u) + c[/tex]

Integrate du using constant rule

[tex]u\ln(u) - u + c = u\ln(u) - f(u) + c[/tex]

Subtract c from both sides

[tex]u\ln(u) - u = u\ln(u) - f(u)[/tex]

Subtract u ln(u) from both sides

[tex]- u = - f(u)[/tex]

Rewrite as:

[tex]f(u) = u[/tex]

Thanks for stopping by. We are committed to providing the best answers for all your questions. See you again soon. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.