Integrals
This section will cover integrals and related topics, the Fundamental Theorem of Calculus, and some other advanced calculus topics.
The there are two types of integrals:
Indefinite Integral
The indefinite integral, or antiderivative, is a partial inverse of the derivative. That is, if the derivative of a function is written as , then the indefinite integral of is , where is a real constant. This is because the derivative of a constant is .
Notation
- The integral of a function
is written as , where the means that the function is being integrated in relation to .
- Often, to save space, the integral of
is written as , the integral of as , etc.
Rules of Indefinite Integrals
for a constant and another constant .
![$\int f(x)+g(x)...+z(x)\,dx=\int f(x)\,dx+\int g(x)\,dx...+\int z(x)\,dx$](//latex.artofproblemsolving.com/e/1/6/e16bea40e04e8f3db71565d95cef3042f3cb1a5f.png)
, ![$n \ne -1$](//latex.artofproblemsolving.com/d/0/a/d0a14bfbe708a6cdf5d6961a5ca917965caba76d.png)
![$\int x^{-1}\,dx=\ln |x|+c$](//latex.artofproblemsolving.com/4/6/4/4641a83c43665970ebd05c134d421de62b5f1ef3.png)
![$\int \sin x\,dx = -\cos x + c$](//latex.artofproblemsolving.com/5/6/1/5612a052d77b988d9a96178ad94ae84f91d46f6e.png)
![$\int \cos x\,dx = \sin x + c$](//latex.artofproblemsolving.com/c/7/7/c77520523ae15b3145e86c2819661bd9afb8a815.png)
![$\int\tan x\,dx = \ln |\cos x| + c$](//latex.artofproblemsolving.com/d/6/f/d6ffa770c25e8f856f04e4cccb6081f9cab6e670.png)
![$\int \sec x\,dx = \ln |\sec x + \tan x| + c$](//latex.artofproblemsolving.com/2/5/0/25063c6b0aa1dca9e9c5311f2f9b38dfc11d716b.png)
![$\int \csc \, dx =\ln |\csc x + \cot x| + c$](//latex.artofproblemsolving.com/f/b/7/fb7386ebd5c389951a584aa33b92130bd465ab34.png)
![$\int \cot x\,dx = \ln |\sin x| + c$](//latex.artofproblemsolving.com/c/5/5/c552f70d7309ac19ad504e5227c002fb8bfe0be1.png)
![$\int cx\, dx=c\int x\,dx$](//latex.artofproblemsolving.com/2/2/d/22db6411a22b2cecc298ca33c558e55ca875ab10.png)
Definite Integral
The definite integral is also the area under a curve between two points and . For example, the area under the curve between and is , as are below the x-axis is taken as negative area.
Definition and Notation
- The definite integral of a function between
and is written as .
, where is the antiderivative of . This is also notated $\int f(x)\,dx \eval^{b}_{a}$ (Error compiling LaTeX. Unknown error_msg), read as "The integral of evaluated at and ." Note that this means in definite integration, one need not add a constant, as the constants from the functions cancel out.
Rules of Definite Integrals
for any .
Fundamental Theorem of Calculus
Let , , . Suppose is differentiable on the whole interval (using limits from the right and left for the derivatives at and , respectively), and suppose that is Riemann integrable on . Then .
In other words, "the total change (on the right) is the sum of all the little changes (on the left)."
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