mrdoe said:Homework Statement
[tex]\displaystyle\int\left(\dfrac{\ln x}{x}\right)^2 dx[/tex]
2. The attempt at a solution
I tried letting [tex]u=\ln^2 x[/tex] and dv the rest and I also tried [tex]dv=\ln^2 x dx[/tex] and u the rest. It won't work out.
The natural logarithm, denoted as ln(x) or loge(x), is a mathematical function that gives the power to which the base number e (approximately 2.71828) must be raised to equal a given number x. It is the inverse function of the exponential function.
The integral with natural logarithm represents the area under the curve of a function that includes the natural logarithm. It is a mathematical operation used to find the antiderivative of a function, which can be used to solve various problems in mathematics and science.
Solving a weird integral with natural logarithm requires using integration techniques such as substitution, integration by parts, or partial fractions. It is important to follow the correct steps and apply the appropriate technique to evaluate the integral.
Integrals with natural logarithms have many applications in fields such as physics, chemistry, economics, and engineering. They are used to model growth and decay, calculate work and energy, and solve optimization problems, among other things.
One example of a weird integral with natural logarithm is ∫ ln(x)/x dx. This integral can be solved using integration by parts, where u = ln(x) and dv = dx/x. The resulting integral can then be evaluated using the logarithmic and inverse tangent functions.