Math / Calculus

QuestionEvaluate the definite integral. $\int_{1}^{2}\left(5+\frac{1}{u}+\frac{1}{u^{2}}\right) d u$

Studdy Solution

STEP 1

What is this asking? We need to find the definite integral of a function with some fractions from $u = 1$ to $u = 2$ . Watch out! Don't forget to plug in both the upper and lower limits of integration and be careful with those fractions!

STEP 2

1. Rewrite the integrand
2. Integrate each term
3. Evaluate the definite integral

STEP 3

Let's rewrite the integrand to make it easier to integrate.
We can rewrite $\frac{1}{u}$ as $u^{-1}$ and $\frac{1}{u^2}$ as $u^{-2}$ .
This makes it clearer how to apply the power rule for integration.
So, our integral becomes: $\int_{1}^{2} (5 + u^{-1} + u^{-2}) \, du$

STEP 4

Now, we'll integrate each term separately.
The integral of $5$ with respect to $u$ is $5u$ .
The integral of $u^{-1}$ is $\ln|u|$ .
The integral of $u^{-2}$ is $\frac{u^{-1}}{-1}$ , which simplifies to $-u^{-1}$ .
Remember, we add one to the exponent and then divide by the new exponent.
Putting it all together, we get: $5u + \ln|u| - u^{-1}$ Don't forget, we still need to evaluate this from $u = 1$ to $u = 2$ !

STEP 5

Let's plug in our limits of integration.
We'll start with the upper limit, $u = 2$ : $5 \cdot 2 + \ln|2| - 2^{-1} = 10 + \ln 2 - \frac{1}{2}$

STEP 6

Now, let's plug in the lower limit, $u = 1$ : $5 \cdot 1 + \ln|1| - 1^{-1} = 5 + 0 - 1 = 4$

STEP 7

Finally, we subtract the result from the lower limit from the result of the upper limit: $(10 + \ln 2 - \frac{1}{2}) - 4 = 6 + \ln 2 - \frac{1}{2} = \frac{11}{2} + \ln 2$

STEP 8

Our final answer is $\frac{11}{2} + \ln 2$ .

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms