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meriva
3 years ago
6

This question is designed to be answered without a calculator. Use the antiderivative formula shown, where f(u) represents a fun

ction. Integral of (l n u) d u = u ln u – f(u) + C Which function represents f(u)?
Mathematics
1 answer:
Phoenix [80]3 years ago
8 0

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

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