Answer :

To solve the integral [tex]\int 3 \ln(x) \, dx[/tex], we'll use integration by parts, a technique that is particularly useful when integrating the product of two functions.

The formula for integration by parts is:

[tex]\int u \, dv = uv - \int v \, du[/tex]

For this integral, we can choose:


  • [tex]u = \ln(x)[/tex] (since the derivative of [tex]\ln(x)[/tex] is simpler)

  • [tex]dv = 3 \, dx[/tex]


Then, we differentiate and integrate to find [tex]du[/tex] and [tex]v[/tex]:


  • [tex]du = \frac{1}{x} \, dx[/tex]

  • [tex]v = 3x[/tex]


Substituting into the integration by parts formula, we get:

[tex]\int 3 \ln(x) \, dx = 3x \ln(x) - \int 3x \cdot \frac{1}{x} \, dx[/tex]

Simplify the remaining integral:

[tex]= 3x \ln(x) - \int 3 \, dx[/tex]

The integral [tex]\int 3 \, dx[/tex] is straightforward:

[tex]\int 3 \, dx = 3x + C[/tex] [tex](C[/tex] is the constant of integration).

Hence, substituting back, we find:

[tex]\int 3 \ln(x) \, dx = 3x \ln(x) - 3x + C[/tex]

Therefore, the correct answer from the given multiple-choice options is 3) [tex]3x \ln(x) - 3x + C[/tex].

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Rewritten by : Barada