Math Snap

STEP 1

1. We are given the integral $\int \frac{1}{v(\ln v - 1)} \, dv$.
2. We need to find the antiderivative of the given function.

STEP 2

1. Simplify the integrand if possible.
2. Use substitution to simplify the integral.
3. Solve the integral using the substitution.
4. Back-substitute to express the solution in terms of the original variable.

STEP 3

Examine the integrand $\frac{1}{v(\ln v - 1)}$ to see if it can be simplified directly. In this case, direct simplification is not possible, so we proceed to substitution.

STEP 4

Use substitution to simplify the integral. Let $u = \ln v - 1$. Then, the derivative $du = \frac{1}{v} \, dv$.

STEP 5

Rewrite the integral in terms of $u$:
$$\int \frac{1}{v(\ln v - 1)} \, dv = \int \frac{1}{u} \cdot v \cdot \frac{1}{v} \, dv = \int \frac{1}{u} \cdot du$$ This simplifies to:
$$\int \frac{1}{u} \, du$$

STEP 6

Solve the integral $\int \frac{1}{u} \, du$:
$$\int \frac{1}{u} \, du = \ln |u| + C$$ where $C$ is the constant of integration.

Back-substitute $u = \ln v - 1$ into the solution:
$$\ln |u| + C = \ln |\ln v - 1| + C$$ The solution to the integral is:
$$\boxed{\ln |\ln v - 1| + C}$$