First make a substitution and then use integration by parts to evaluate the integral. (Remember the constant of integration.)

 x ln(3 + x) dx

To solve the integral \( \int x \ln(3 + x) \, dx \), we'll first make a substitution, and then apply integration by parts.

### Step 1: Substitution
Let’s set:
\[
u = 3 + x
\]
This implies:
\[
du = dx
\]
Also, \( x = u - 3 \), so the integral becomes:
\[
\int (u - 3) \ln(u) \, du
\]
We can now split the integral into two parts:
\[
\int u \ln(u) \, du - 3 \int \ln(u) \, du
\]

### Step 2: Integration by Parts for Each Integral

#### First Integral: \( \int u \ln(u) \, du \)
To solve \( \int u \ln(u) \, du \), we use integration by parts:
- Let \( v = \ln(u) \), so \( dv = \frac{1}{u} \, du \).
- Let \( dw = u \, du \), so \( w = \frac{u^2}{2} \).

Now, applying the integration by parts formula \( \int v \, dw = v w - \int w \, dv \), we get:
\[
\int u \ln(u) \, du = \frac{u^2}{2} \ln(u) - \int \frac{u^2}{2} \cdot \frac{1}{u} \, du
\]
\[
= \frac{u^2}{2} \ln(u) - \int \frac{u}{2} \, du
\]
\[
= \frac{u^2}{2} \ln(u) - \frac{u^2}{4}
\]

#### Second Integral: \( -3 \int \ln(u) \, du \)
We use integration by parts again:
- Let \( v = \ln(u) \), so \( dv = \frac{1}{u} \, du \).
- Let \( dw = du \), so \( w = u \).

Thus:
\[
\int \ln(u) \, du = u \ln(u) - u
\]
Multiplying by \(-3\), we get:
\[
-3 \int \ln(u) \, du = -3(u \ln(u) - u)
= -3u \ln(u) + 3u
\]

### Step 3: Combine the Results
Now, adding both parts, the integral becomes:
\[
\frac{u^2}{2} \ln(u) - \frac{u^2}{4} - 3u \ln(u) + 3u + C
\]
Finally, substitute \( u = 3 + x \) back:
\[
= \frac{(3 + x)^2}{2} \ln(3 + x) - \frac{(3 + x)^2}{4} - 3(3 + x) \ln(3 + x) + 3(3 + x) + C
\]

This is the evaluated integral.

Would you like further clarification or details on any specific part?

### Related Questions:
1. How is integration by parts derived from the product rule?
2. Can integration by parts be applied to trigonometric functions?
3. What are the applications of substitution in integral calculus?
4. How do you handle integrals involving logarithmic functions?
5. What are some common errors to avoid in integration by parts?

### Tip:
Always double-check your substitution and integration by parts setup to avoid calculation errors later in the process.

First make a substitution and then use integration by parts to evaluate the integral. (Remember the constant of integration.) x ln(3 + x) dx

Explore how to solve the integral of x ln(3 + x) using substitution and integration by parts. This calculus problem involves breaking down the integral into manageable parts and applying the integration by parts theorem step by step. Ideal for undergraduate calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation