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Calculus Volume 1

Key Equations

Calculus Volume 1Key Equations

Key Equations

Properties of Sigma Notation i=1nc=nci=1nc=nc
Sums and Powers of Integers i=1ni=1+2++n=n(n+1)2i=1ni=1+2++n=n(n+1)2
Left-Endpoint Approximation ALn=f(x0)Δx+f(x1)Δx++f(xn1)Δx=i=1nf(xi1)ΔxALn=f(x0)Δx+f(x1)Δx++f(xn1)Δx=i=1nf(xi1)Δx
Right-Endpoint Approximation ARn=f(x1)Δx+f(x2)Δx++f(xn)Δx=i=1nf(xi)ΔxARn=f(x1)Δx+f(x2)Δx++f(xn)Δx=i=1nf(xi)Δx
Definite Integral abf(x)dx=limni=1nf(xi*)Δxabf(x)dx=limni=1nf(xi*)Δx
Properties of the Definite Integral aaf(x)dx=0aaf(x)dx=0
abcf(x)dx=cabf(x)abcf(x)dx=cabf(x) for constant c
Mean Value Theorem for Integrals If f(x)f(x) is continuous over an interval [a,b],[a,b], then there is at least one point c[a,b]c[a,b] such that f(c)=1baabf(x)dx.f(c)=1baabf(x)dx.
Fundamental Theorem of Calculus Part 1 If f(x)f(x) is continuous over an interval [a,b],[a,b], and the function F(x)F(x) is defined by F(x)=axf(t)dt,F(x)=axf(t)dt, then F(x)=f(x).F(x)=f(x).
Fundamental Theorem of Calculus Part 2 If f is continuous over the interval [a,b][a,b] and F(x)F(x) is any antiderivative of f(x),f(x), then abf(x)dx=F(b)F(a).abf(x)dx=F(b)F(a).
Net Change Theorem F(b)=F(a)+abF'(x)dxF(b)=F(a)+abF'(x)dx or abF'(x)dx=F(b)F(a)abF'(x)dx=F(b)F(a)
Substitution with Indefinite Integrals f[g(x)]g(x)dx=f(u)du=F(u)+C=F(g(x))+Cf[g(x)]g(x)dx=f(u)du=F(u)+C=F(g(x))+C
Substitution with Definite Integrals abf(g(x))g'(x)dx=g(a)g(b)f(u)duabf(g(x))g'(x)dx=g(a)g(b)f(u)du
Integrals of Exponential Functions exdx=ex+Cexdx=ex+C
Integration Formulas Involving Logarithmic Functions x−1dx=ln|x|+Cx−1dx=ln|x|+C
Integrals That Produce Inverse Trigonometric Functions dua2u2=sin−1(ua)+Cdua2u2=sin−1(ua)+C
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