Find \$\int \frac{2}{x} d x\$
Answer (2)
The integral ∫ x 2 d x can be computed using the constant multiple rule. The result is 2 ln ∣ x ∣ + C , where C is the constant of integration. This method involves recognizing that the integral of x 1 is ln ∣ x ∣ + C .
;
Apply the constant multiple rule: ∫ x 2 d x = 2 ∫ x 1 d x .
Integrate x 1 to get ln ∣ x ∣ .
Multiply by the constant: 2 ln ∣ x ∣ .
Add the constant of integration: 2 ln ∣ x ∣ + C
Explanation
Problem Analysis We are asked to find the indefinite integral of the function x 2 with respect to x .
Applying Constant Multiple Rule We can use the constant multiple rule for integration, which states that ∫ c f ( x ) d x = c ∫ f ( x ) d x , where c is a constant. In our case, c = 2 and f ( x ) = x 1 . So, we can rewrite the integral as:
Rewriting the Integral ∫ x 2 d x = 2 ∫ x 1 d x
Integrating 1/x We know that the indefinite integral of x 1 is ln ∣ x ∣ + C , where C is the constant of integration. Therefore, we have:
Substituting the Integral 2 ∫ x 1 d x = 2 ( ln ∣ x ∣ + C ) = 2 ln ∣ x ∣ + 2 C
Simplifying the Expression Since 2 C is also an arbitrary constant, we can replace it with C . Thus, the final answer is:
Final Answer 2 ln ∣ x ∣ + C
Examples
In electrical engineering, when analyzing circuits, you might encounter integrals of the form ∫ t I d t , where I is current and t is time. Solving such integrals helps determine the charge accumulation over time. Similarly, in physics, calculating the work done by a force that varies inversely with distance involves integrating functions like x k , where k is a constant and x is distance. Understanding how to integrate these functions is crucial for solving real-world problems in both fields.
