3.6 Q-13

Question Statement

Evaluate the integral: $\int_{1}^{2} \ln x , dx$

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Background and Explanation

This is an integral involving the natural logarithm function, $\ln x$. To solve this, we can use integration by parts, a technique based on the product rule of differentiation. The formula for integration by parts is: $\int u , dv = uv - \int v , du$

In this case, we’ll choose $u = \ln x$ and $dv = dx$, and then differentiate and integrate as required.

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Solution

1. Set Up Integration by Parts:

   Using the formula for integration by parts, let: $u = \ln x \quad \text{and} \quad dv = dx$

   Now, differentiate and integrate: $du = \frac{1}{x} , dx \quad \text{and} \quad v = x$

2. Apply the Formula:

   Using the integration by parts formula: $\int u , dv = uv - \int v , du$

   Substitute the values of $u$, $v$, and $du$ into the equation: $\int \ln x , dx = x \ln x - \int x \times \frac{1}{x} , dx$

   Simplify the second integral: $\int \ln x , dx = x \ln x - \int 1 , dx$

3. Evaluate the Integrals:

   The first term is straightforward: $x \ln x \Big|_1^2 = 2 \ln 2 - 1 \ln 1$

   Since $\ln 1 = 0$, the second term becomes: $2 \ln 2 - 0 = 2 \ln 2$

   Now, evaluate the second integral: $\int_1^2 1 , dx = x \Big|_1^2 = 2 - 1 = 1$

4. Final Calculation:

   Now, combine the results: $2 \ln 2 - 1$

   Thus, the value of the integral is: $\boxed{2 \ln 2 - 1}$

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Key Formulas or Methods Used

• Integration by Parts: $\int u , dv = uv - \int v , du$

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Summary of Steps

1. Apply integration by parts with $u = \ln x$ and $dv = dx$, leading to: $\int \ln x , dx = x \ln x - \int 1 , dx$

2. Evaluate $x \ln x$ at the limits: $2 \ln 2 - 1 \ln 1 = 2 \ln 2$

3. Evaluate the integral $\int_1^2 1 , dx$ to get $1$.

4. Combine the results to get the final answer: $2 \ln 2 - 1$