3.3 Q-4

Question Statement

Evaluate the following integral:

$$\int \frac{1}{x \ln x} , dx$$

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Background and Explanation

This problem involves an integral that has a logarithmic expression both in the denominator and inside the logarithm. To solve it, we will use the substitution method. Specifically, we’ll let the inner logarithmic function be a new variable, which will simplify the integral.

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Solution

1. Substitute for simplification:
   We start by making the substitution:

$$u = \ln x$$

Differentiating both sides with respect to $x$:

$$\frac{du}{dx} = \frac{1}{x} \quad \text{or} \quad dx = x , du$$

2. Rewrite the integral:
   Substituting $u = \ln x$ and $dx = x , du$ into the original integral, we get:

$$\int \frac{1}{x \ln x} , dx = \int \frac{1}{x} \cdot \frac{1}{\ln x} , dx = \int \frac{1}{u} , du$$

3. Solve the simplified integral:
   The integral of $\frac{1}{u}$ is a standard form:

$$\int \frac{1}{u} , du = \ln |u| + C$$

Substituting back $u = \ln x$:

$$\ln |\ln x| + C$$

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Key Formulas or Methods Used

• Substitution:
  The substitution $u = \ln x$ was used to simplify the integral.

• Standard Integral:
  The integral of $\frac{1}{u}$ is:

$$\int \frac{1}{u} , du = \ln |u| + C$$

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Summary of Steps

1. Make the substitution $u = \ln x$.
2. Substitute into the integral, giving $\int \frac{1}{u} , du$.
3. Integrate $\frac{1}{u}$ to get $\ln |u| + C$.
4. Substitute back $u = \ln x$ to get the final answer:
   $\ln |\ln x| + C$