The antiderivative of ln(x), also known as the natural logarithm of x, is a fundamental concept in calculus. The natural logarithm function, denoted as ln(x), is the inverse of the exponential function e^x. To find the antiderivative of ln(x), we can use integration by parts, which is a technique used to integrate products of functions.
Integration by parts states that the integral of a product of two functions u and v can be expressed as the integral of u times the derivative of v, minus the integral of the derivative of u times v. Mathematically, this can be represented as ∫u dv = uv - ∫v du. In the case of the antiderivative of ln(x), we can choose u = ln(x) and dv = dx, which implies du = 1/x dx and v = x.
Key Points
- The antiderivative of ln(x) can be found using integration by parts.
- The formula for integration by parts is ∫u dv = uv - ∫v du.
- We can choose u = ln(x) and dv = dx to find the antiderivative of ln(x).
- The antiderivative of ln(x) is x*ln(x) - x + C, where C is the constant of integration.
- The antiderivative of ln(x) has numerous applications in calculus, including finding the area under curves and solving differential equations.
Derivation of the Antiderivative of ln(x)
Using the formula for integration by parts, we can derive the antiderivative of ln(x) as follows: ∫ln(x) dx = x*ln(x) - ∫x * 1/x dx. Simplifying the integral on the right-hand side, we get ∫ln(x) dx = x*ln(x) - ∫1 dx. The integral of 1 with respect to x is x, so we have ∫ln(x) dx = x*ln(x) - x + C, where C is the constant of integration.
Applications of the Antiderivative of ln(x)
The antiderivative of ln(x) has numerous applications in calculus, including finding the area under curves and solving differential equations. For example, the antiderivative of ln(x) can be used to find the area under the curve y = ln(x) between two points, say a and b. This can be done by evaluating the definite integral ∫[a,b] ln(x) dx, which gives us the area under the curve between the two points.
| Function | Antiderivative |
|---|---|
| ln(x) | x*ln(x) - x + C |
| 1/x | ln|x| + C |
| e^x | e^x + C |
Technical Specifications and Contextual Explanation
The antiderivative of ln(x) is a continuous function on the interval (0, ∞) and has a vertical asymptote at x = 0. The function is also differentiable on the interval (0, ∞) and has a derivative equal to 1/x. The antiderivative of ln(x) can be used to model various real-world phenomena, such as population growth, chemical reactions, and electrical circuits.
The antiderivative of ln(x) can also be used to solve differential equations, which are equations that involve an unknown function and its derivatives. For example, the differential equation dy/dx = 1/x can be solved using the antiderivative of ln(x), which gives us the general solution y = ln|x| + C.
Balance of Technical Precision and Accessibility
The antiderivative of ln(x) is a complex concept that requires a balance of technical precision and accessibility. While the technical details of the antiderivative are important, it is also essential to provide an accessible explanation that can be understood by students and non-experts. By using clear and concise language, providing examples and illustrations, and avoiding unnecessary technical jargon, we can make the antiderivative of ln(x) more accessible and easier to understand.
What is the antiderivative of ln(x)?
+The antiderivative of ln(x) is x*ln(x) - x + C, where C is the constant of integration.
How is the antiderivative of ln(x) derived?
+The antiderivative of ln(x) is derived using integration by parts, with u = ln(x) and dv = dx.
What are the applications of the antiderivative of ln(x)?
+The antiderivative of ln(x) has numerous applications in calculus, including finding the area under curves and solving differential equations.
