Math Snap

STEP 1

1. We are given the integral $\int \frac{\ln(x)}{x} \, \mathrm{dx}$.
2. We need to perform a $u$-substitution to simplify the integral.
3. The goal is to express the integral in terms of $u$.

STEP 2

1. Choose an appropriate substitution for $u$.
2. Differentiate the chosen substitution to find $du$.
3. Substitute $u$ and $du$ into the integral.
4. Simplify the integral in terms of $u$.

STEP 3

Choose an appropriate substitution. Let $u = \ln(x)$.

STEP 4

Differentiate the substitution to find $du$:
$$\frac{du}{dx} = \frac{1}{x}$$ Thus,
$$du = \frac{1}{x} \, dx$$

STEP 5

Substitute $u$ and $du$ into the integral:
The original integral is:
$$\int \frac{\ln(x)}{x} \, \mathrm{dx}$$ Substituting $u = \ln(x)$ and $du = \frac{1}{x} \, dx$, we get:
$$\int u \, du$$

The new integral in terms of $u$ is:
$$\int u \, du$$ This matches the option $\int u \, du$.