You should already have seen the substitution method, which is one of the first techniques that are presented. Also, it's also one that you should try first when you have an integration problem. This technique might not be useful in some cases, but it's reasonably simple, so if it doesn't work, you haven't wasted much time.5P@N said:If you could just show me a couple of relevant integration rules
The integral of x/(1+x^2)dx is equal to 1/2ln(1+x^2) because of the inverse relationship between the natural logarithm and the derivative of the inverse hyperbolic tangent function, which is the antiderivative of x/(1+x^2). This relationship is based on the fundamental theorem of calculus.
The constant 1/2 in the integral of x/(1+x^2)dx represents the relationship between the inverse hyperbolic tangent function and the natural logarithm. It is derived from the integral of the derivative of the inverse hyperbolic tangent function, which is equal to 1/2ln(1+x^2).
The antiderivative of x/(1+x^2)dx is determined using integration techniques such as substitution and integration by parts. In this case, the inverse hyperbolic tangent function is used as the substitution to simplify the integral. The result is then evaluated and simplified to 1/2ln(1+x^2).
The proof of ∫x/(1+x^2)dx = 1/2ln(1+x^2) is based on the fundamental theorem of calculus and the properties of the inverse hyperbolic tangent function and the natural logarithm. By taking the derivative of 1/2ln(1+x^2), the result will be x/(1+x^2). This shows that the two are inverses of each other, and thus the integral is equal to the antiderivative.
The integral of x/(1+x^2)dx is used in various fields such as physics, engineering, and economics to solve problems involving rates of change, optimization, and area under curves. It is also used in the process of finding the inverse of a function. Additionally, the integral has applications in probability and statistics, such as in the calculation of expected values.