Consider the following.
9

x
 ln(x) dx

The integrand 
9

x
 ln(x)
 can be rewritten as the product of which two functions?


(ln(x))−1 and 9x1/2

(ln(x))−1 and 9x−1/2
    

ln(x) and 9x1/2

ln(x) and 9x−1/2
Correct: Your answer is correct.
Outstanding!
Which of the two functions is most easily integrated?
  
Correct: Your answer is correct.
 That's right!
Use integration by parts to evaluate the integral. (Remember the constant of integration.)
  
Incorrect: Your answer is incorrect.

Check your answer by taking the derivative.

To solve the integral \( \int 9x \ln(x) \, dx \) using **integration by parts**, we need to select two functions \( u \) and \( dv \) such that:

\[
\int u \, dv = u v - \int v \, du
\]

We can break down the integrand \( 9x \ln(x) \) into a product of two functions:

- \( u = \ln(x) \)
- \( dv = 9x \, dx \)

### Step 1: Differentiate and integrate
Now, we compute \( du \) and \( v \):

1. \( u = \ln(x) \implies du = \frac{1}{x} \, dx \)
2. \( dv = 9x \, dx \implies v = \frac{9}{2}x^2 \)

### Step 2: Apply the integration by parts formula
Now substitute into the formula:

\[
\int 9x \ln(x) \, dx = \ln(x) \cdot \frac{9}{2}x^2 - \int \frac{9}{2}x^2 \cdot \frac{1}{x} \, dx
\]

Simplify the second integral:

\[
\int 9x \ln(x) \, dx = \frac{9}{2}x^2 \ln(x) - \frac{9}{2} \int x \, dx
\]

### Step 3: Solve the remaining integral
\[
\frac{9}{2} \int x \, dx = \frac{9}{2} \cdot \frac{x^2}{2} = \frac{9}{4}x^2
\]

### Step 4: Combine terms
Now, substitute back:

\[
\int 9x \ln(x) \, dx = \frac{9}{2}x^2 \ln(x) - \frac{9}{4}x^2 + C
\]

Thus, the evaluated integral is:

\[
\int 9x \ln(x) \, dx = \frac{9}{2}x^2 \ln(x) - \frac{9}{4}x^2 + C
\]

### Step 5: Check the result by differentiating
Differentiate the result to confirm:

\[
\frac{d}{dx} \left( \frac{9}{2}x^2 \ln(x) - \frac{9}{4}x^2 \right)
\]

Using the product rule on \( \frac{9}{2}x^2 \ln(x) \):

\[
\frac{d}{dx} \left( \frac{9}{2}x^2 \ln(x) \right) = \frac{9}{2} \left( 2x \ln(x) + x \right) = 9x \ln(x) + \frac{9}{2}x
\]

Now differentiate \( -\frac{9}{4}x^2 \):

\[
\frac{d}{dx} \left( -\frac{9}{4}x^2 \right) = -\frac{9}{2}x
\]

Add the two results:

\[
9x \ln(x) + \frac{9}{2}x - \frac{9}{2}x = 9x \ln(x)
\]

Since the derivative matches the original integrand, the solution is correct.

### Questions:
1. Can you explain why integration by parts works in this case?
2. What happens if we choose different functions for \( u \) and \( dv \)?
3. How do you decide which function to assign to \( u \) and \( dv \) in integration by parts?
4. Can we evaluate this integral using another method, such as substitution?
5. How does the constant of integration play a role in indefinite integrals?

**Tip:** When applying integration by parts, choosing the correct \( u \) and \( dv \) can simplify the integration process significantly. Look for functions that become simpler after differentiation.

Consider the following.
9x ln(x) dx
The integrand 9x ln(x) can be rewritten as the product of which two functions?
(ln(x))−1 and 9x1/2
(ln(x))−1 and 9x−1/2
ln(x) and 9x1/2
ln(x) and 9x−1/2
Correct: Your answer is correct. Outstanding! Which of the two functions is most easily integrated?
Correct: Your answer is correct. That's right! Use integration by parts to evaluate the integral. (Remember the constant of integration.)
Incorrect: Your answer is incorrect. Check your answer by taking the derivative.

Learn how to evaluate the integral of 9x ln(x) using the method of integration by parts. This detailed solution walks you through the process step-by-step, covering key calculus concepts such as logarithmic functions and polynomial integration. Suitable for undergraduate calculus students.

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