integrate 3lnx
Understand the Problem
The question is asking to find the integral of the function 3ln(x). The integral will involve applying integration techniques and rules for logarithmic functions.
Answer
$$ \int 3\ln(x) \, dx = 3x \ln(x)  3x + C $$
Answer for screen readers
The integral of the function $3\ln(x)$ is given by: $$ \int 3\ln(x) , dx = 3x \ln(x)  3x + C $$
Steps to Solve

Identify the integral We need to find the integral of the function $3\ln(x)$, which can be written as: $$ \int 3\ln(x) , dx $$

Factor out constant Since $3$ is a constant, we can factor it out of the integral: $$ 3 \int \ln(x) , dx $$

Use integration by parts To solve $\int \ln(x) , dx$, we will use integration by parts. We set:
 $u = \ln(x)$, then $du = \frac{1}{x} dx$
 $dv = dx$, then $v = x$
Now, we can apply the integration by parts formula: $$ \int u , dv = uv  \int v , du $$

Apply the formula Substituting our values into the integration by parts formula: $$ \int \ln(x) , dx = x \ln(x)  \int x \cdot \frac{1}{x} , dx $$ This simplifies to: $$ \int \ln(x) , dx = x \ln(x)  \int 1 , dx $$ Thus we continue to solve the integral: $$ \int \ln(x) , dx = x \ln(x)  x + C $$

Combine results Now returning to our original integral, $$ \int 3\ln(x) , dx = 3 \left( x \ln(x)  x + C \right) $$

Final expression Thus, we have: $$ \int 3\ln(x) , dx = 3x \ln(x)  3x + C $$ where $C$ is the constant of integration.
The integral of the function $3\ln(x)$ is given by: $$ \int 3\ln(x) , dx = 3x \ln(x)  3x + C $$
More Information
The integral of the logarithmic function is a common application in calculus, especially in problems involving growth rates and areas under curves. Using integration by parts is a key technique that allows for solving integrals of products involving logarithmic and polynomial functions.
Tips
 Forgetting to apply integration by parts correctly or misidentifying $u$ and $dv$ can lead to incorrect results. Itâ€™s important to choose substitutions that simplify the integral.
 Omitting the constant of integration $C$ at the end can result in incomplete answers.