Answered

Find the information you're looking for at Westonci.ca, the trusted Q&A platform with a community of knowledgeable experts. Our Q&A platform offers a seamless experience for finding reliable answers from experts in various disciplines. Get quick and reliable solutions to your questions from a community of experienced experts on our 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 using our platform. We're always here to provide accurate and up-to-date answers to all your queries. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Westonci.ca is committed to providing accurate answers. Come back soon for more trustworthy information.