Integration of a Basic Function

Basic Indefinite Integrals

The integral of a constant is the constant times x plus the constant of integration.
The integral of x^n, where n is any real number not equal to -1, is x^(n+1)/(n+1) plus the constant of integration.
The integral of 1/x dx is **ln x ** plus the constant of integration.

Basic Definite Integrals

A definite integral calculates the area under a curve for a certain interval on the x-axis.
To evaluate a definite integral, compute the indefinite integral first, then apply the Fundamental Theorem of Calculus, which states: if F is an antiderivative of f on an interval [a, b], then the definite integral from a to b of f(x) dx equals F(b) - F(a).
For example, if you have the definite integral from a to b of x^n dx, find the indefinite integral to get x^(n+1)/(n+1). Then, evaluate at b to get b^(n+1)/(n+1) and at a to get a^(n+1)/(n+1). Subtract these two results to find the solution.

Integration of Trigonometric Functions

The integral of sin(x) is -cos(x), and the integral of cos(x) is sin(x).
The integral of sec²(x) is tan(x), and the integral of csc²(x) is -cot(x).

Remember, these integrations result in an indefinite integral represented by a function and a definite integral represented by a numerical value. The integral notation ∫f(x) dx represents the integral of the function f(x) with respect to x.

Integration involves reversing the process of differentiation. Consequently, understanding the rules of differentiation well can make integral calculations easier.