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

$$ \int 3\ln(x) \, dx = 3x \ln(x) - 3x + C $$

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. It’s important to choose substitutions that simplify the integral.
Omitting the constant of integration $C$ at the end can result in incomplete answers.

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