Antiderivatives are the opposite of derivatives. An antiderivative is a function that reverses what the derivative does. One function has many antiderivatives, but they all take the form of a function plus an arbitrary constant. Antiderivatives are a key part of indefinite integrals.

 Indefinite Integral
An indefinite integral is a function that takes the antiderivative of another function. It is visually represented as an integral symbol, a function, and then a dx at the end. The indefinite integral is an easier way to symbolize taking the antiderivative. The indefinite integral is related to the definite integral, but the two are not the same.

 Complicated Indefinite Integrals
Not all indefinite integrals follow one simple rule. Some are slightly more complicated, but they can be made easier by remembering the derivatives they came from. These complicated indefinite integrals include the integral of a constant (the constant times x), the integral of e^x (e^x) and the integral of x^-1 (ln[x]).

 Simple Substitutions
The substitution method is useful on some indefinite integrals that are not as simple as they look. These include functions such as e^(5-2x) or the square root of [x-3], in which the variable part of the function is more complex than just an x. These simple substitutions require use of the substitution method to solve.

 Tricky Substitutions Involving Radicals
When an indefinite integral involves a function of x times the radical of another function of x, the substitution method must be used in multiple ways. In these tricky substitutions involving radicals, the dx and the x's in both parts of the problem must be substituted for to in order to solve the problems.

 Velocity and Acceleration
If position is given by a function p(x), then the velocity is the first derivative of that function, and the acceleration is the second derivative. By using differential equations with either velocity or acceleration, it is possible to find position and velocity functions from a known acceleration.

Finding the antiderivatives of a function require a little backwards thinking. Since the derivative of the wanted antiderivative is the given function, checking for correctness is easy. You just take the derivative, and see if it is the given function. Also, antiderivatives of functions happen to be not just one function, but a whole family of functions. This family can be written as a polynomial plus c, where c stands for any constant.

 Indefinite Integrals
Indefinite integrals are functions that do the opposite of what derivatives do. They represent taking the antiderivatives of functions. A formula useful for solving indefinite integrals is that the integral of x to the nth power is one divided by n+1 times x to the n+1 power, all plus a constant term.

 Substitution Method
The substitution method is a very valuable way to evaluate some indefinite integrals. The substitution method adds a new function into the one being integrated, and substitutes the new function and its derivative in order to make finding the wanted antiderivative easier.

 Natural Log Integration
Because the derivatives of e^x and ln(x) are e^x and 1/x, respectively, the integrals of the latter two functions are just the former. If the variable parts of these functions are not just x, but a polynomial of x, then the substitution method must be used to get the correct answer.

 Differential Equations
Differential equations are equations with a derivative of an unknown function. Solving a differential equation requires using antidifferentiation. Since they use antiderivatives, there are multiple solutions. Differential equations can be classified by their order, which is the same as the largest derivative in the equation (1st, 2nd, etc.).

 Differential Equation Model
If a function is growing or shrinking exponentially, it can be modeled using a differential equation. The equation itself is dy/dx=ky, which leads to the solution of y=ce^(kx). In the differential equation model, k is a constant that determines if the function is growing or shrinking. If k is greater than 1, the function is growing. If it is less than 1, the function is shrinking.




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