Calculus Volume 1

# Key Equations

Calculus Volume 1Key Equations
• Properties of Sigma Notation
$∑i=1nc=nc∑i=1nc=nc$
$∑i=1ncai=c∑i=1nai∑i=1ncai=c∑i=1nai$
$∑i=1n(ai+bi)=∑i=1nai+∑i=1nbi∑i=1n(ai+bi)=∑i=1nai+∑i=1nbi$
$∑i=1n(ai−bi)=∑i=1nai−∑i=1nbi∑i=1n(ai−bi)=∑i=1nai−∑i=1nbi$
$∑i=1nai=∑i=1mai+∑i=m+1nai∑i=1nai=∑i=1mai+∑i=m+1nai$
• Sums and Powers of Integers
$∑i=1ni=1+2+⋯+n=n(n+1)2∑i=1ni=1+2+⋯+n=n(n+1)2$
$∑i=1ni2=12+22+⋯+n2=n(n+1)(2n+1)6∑i=1ni2=12+22+⋯+n2=n(n+1)(2n+1)6$
$∑i=0ni3=13+23+⋯+n3=n2(n+1)24∑i=0ni3=13+23+⋯+n3=n2(n+1)24$
• Left-Endpoint Approximation
$A≈Ln=f(x0)Δx+f(x1)Δx+⋯+f(xn−1)Δx=∑i=1nf(xi−1)ΔxA≈Ln=f(x0)Δx+f(x1)Δx+⋯+f(xn−1)Δx=∑i=1nf(xi−1)Δx$
• Right-Endpoint Approximation
$A≈Rn=f(x1)Δx+f(x2)Δx+⋯+f(xn)Δx=∑i=1nf(xi)ΔxA≈Rn=f(x1)Δx+f(x2)Δx+⋯+f(xn)Δx=∑i=1nf(xi)Δx$
• Definite Integral
$∫abf(x)dx=limn→∞∑i=1nf(xi*)Δx∫abf(x)dx=limn→∞∑i=1nf(xi*)Δx$
• Properties of the Definite Integral
$∫aaf(x)dx=0∫aaf(x)dx=0$
$∫baf(x)dx=−∫abf(x)dx∫baf(x)dx=−∫abf(x)dx$
$∫ab[f(x)+g(x)]dx=∫abf(x)dx+∫abg(x)dx∫ab[f(x)+g(x)]dx=∫abf(x)dx+∫abg(x)dx$
$∫ab[f(x)−g(x)]dx=∫abf(x)dx−∫abg(x)dx∫ab[f(x)−g(x)]dx=∫abf(x)dx−∫abg(x)dx$
$∫abcf(x)dx=c∫abf(x)∫abcf(x)dx=c∫abf(x)$ for constant c
$∫abf(x)dx=∫acf(x)dx+∫cbf(x)dx∫abf(x)dx=∫acf(x)dx+∫cbf(x)dx$
• 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 $f(c)=1b−a∫abf(x)dx.f(c)=1b−a∫abf(x)dx.$
• 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
$∫f[g(x)]g′(x)dx=∫f(u)du=F(u)+C=F(g(x))+C∫f[g(x)]g′(x)dx=∫f(u)du=F(u)+C=F(g(x))+C$
• Substitution with Definite Integrals
$∫abf(g(x))g'(x)dx=∫g(a)g(b)f(u)du∫abf(g(x))g'(x)dx=∫g(a)g(b)f(u)du$
• Integrals of Exponential Functions
$∫exdx=ex+C∫exdx=ex+C$
$∫axdx=axlna+C∫axdx=axlna+C$
• Integration Formulas Involving Logarithmic Functions
$∫x−1dx=ln|x|+C∫x−1dx=ln|x|+C$
$∫lnxdx=xlnx−x+C=x(lnx−1)+C∫lnxdx=xlnx−x+C=x(lnx−1)+C$
$∫logaxdx=xlna(lnx−1)+C∫logaxdx=xlna(lnx−1)+C$
• Integrals That Produce Inverse Trigonometric Functions
$∫dua2−u2=sin−1(ua)+C∫dua2−u2=sin−1(ua)+C$
$∫dua2+u2=1atan−1(ua)+C∫dua2+u2=1atan−1(ua)+C$
$∫duuu2−a2=1asec−1(ua)+C∫duuu2−a2=1asec−1(ua)+C$