Question

Evaluate the integral. $$\int_{1}^{2} \frac{\ln x}{x} d x$$

   Evaluate the integral.
$$\int_{1}^{2} \frac{\ln x}{x} d x$$

Calculus: Early Transcendental Functions

Calculus: Early Transcendental Functions

Robert T. Smith,… 3rd Edition

Chapter 4, Problem 28

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Solved by Expert Linh Vu

06/04/2020

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Then, we find the derivative of $u$ with respect to $x$, which is $du/dx = 1/x$. Show more…

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Evaluate the integral. $$\int_{1}^{2} \frac{\ln x}{x} d x$$

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Key Concepts

Integration by Substitution

This is a technique where a new variable is introduced to simplify the integral. In problems involving the natural logarithm function, setting u equal to ln x often transforms the integral into an easier form to integrate. The differential and limits of integration are adjusted accordingly, streamlining the evaluation process.

Definite Integrals and Limit Adjustment

When evaluating definite integrals, the limits of integration must be modified to correspond to the new variable after substitution. This ensures that the integral is computed correctly over the transformed interval rather than reverting back to the original variable, preserving the equality of the original integral.

Power Rule of Integration

The power rule is employed to integrate functions of the form u^n, where n is a constant. After substitution transforms the original integral into one involving u raised to a power, the power rule provides a straightforward antiderivative, which can then be evaluated using the adjusted limits to obtain the final result.

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