Question

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)?

Answer 1

Answer:

$f(u) = u$

Step-by-step explanation:

Given

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

Required

Find f(u)

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

$\int {\ln(u)} \, du = u\ln(u) - f(u)+ c$

Using integration by parts, we have:

$\int {fg'} = fg - \int f'g$

So, we have:

$f = \ln(u)$

Differentiate

$f'=\frac{1}{u}$

$g' = 1$

Integrate

$g=u$

So:

$\int {fg'} = fg - \int f'g$

$\int \ln(u)\ du = \ln(u) * u - \int \frac{1}{u} * u\ du$

$\int \ln(u)\ du = u\ln(u) - \int \ du$

So, we have:

$u\ln(u) - \int \ du = u\ln(u) - f(u) + c$

Integrate du using constant rule

$u\ln(u) - u + c = u\ln(u) - f(u) + c$

Subtract c from both sides

$u\ln(u) - u = u\ln(u) - f(u)$

Subtract u ln(u) from both sides

$- u = - f(u)$

Rewrite as:

$f(u) = u$