What is the integral of ln(x) dx?

What is the integral of ln(x) dx?

• A. xln(x) - x + C
• B. x^2/2 + C
• C. ln(x^2)/2 + C
• D. x^2ln(x) + C

Answer: (A)

To find the integral of \( \ln(x) \, dx \), we can use integration by parts. This technique is particularly useful when integrating the product of two functions, and in this case, we can choose: - Let \( u = \ln(x) \) so that \( du = \frac{1}{x} \, dx \). - Let \( dv = dx \) so that \( v = x \). Using the integration by parts formula, which states that \( \int u \, dv = uv - \int v \, du \), we can proceed with the substitution: 1. Calculate \( uv \): \[ uv = x \ln(x) \] 2. Calculate \( \int v \, du \): \[ \int v \, du = \int x \cdot \frac{1}{x} \, dx = \int 1 \, dx = x \] Putting it all together: \[ \int \ln(x) \, dx = x \ln(x) - \int 1 \, dx = x \ln(x) - x + C \] Thus, the final result of the integral is

To find the integral of ( \ln(x) , dx ), we can use integration by parts. This technique is particularly useful when integrating the product of two functions, and in this case, we can choose:

• Let ( u = \ln(x) ) so that ( du = \frac{1}{x} , dx ).

• Let ( dv = dx ) so that ( v = x ).

Using the integration by parts formula, which states that ( \int u , dv = uv - \int v , du ), we can proceed with the substitution:

1. Calculate ( uv ):

[

uv = x \ln(x)

]

2. Calculate ( \int v , du ):

[

\int v , du = \int x \cdot \frac{1}{x} , dx = \int 1 , dx = x

]

Putting it all together:

[

\int \ln(x) , dx = x \ln(x) - \int 1 , dx = x \ln(x) - x + C

]

Thus, the final result of the integral is