Calculus Volume 2

# Key Equations

Calculus Volume 2Key Equations

### Key Equations

 Integration by parts formula $∫udv=uv−∫vdu∫udv=uv−∫vdu$ Integration by parts for definite integrals $∫abudv=uv|ab−∫abvdu∫abudv=uv|ab−∫abvdu$

To integrate products involving $sin(ax),sin(ax),$ $sin(bx),sin(bx),$ $cos(ax),cos(ax),$ and $cos(bx),cos(bx),$ use the substitutions.

 Sine Products $sin(ax)sin(bx)=12cos((a−b)x)−12cos((a+b)x)sin(ax)sin(bx)=12cos((a−b)x)−12cos((a+b)x)$ Sine and Cosine Products $sin(ax)cos(bx)=12sin((a−b)x)+12sin((a+b)x)sin(ax)cos(bx)=12sin((a−b)x)+12sin((a+b)x)$ Cosine Products $cos(ax)cos(bx)=12cos((a−b)x)+12cos((a+b)x)cos(ax)cos(bx)=12cos((a−b)x)+12cos((a+b)x)$ Power Reduction Formula Power Reduction Formula $∫tannxdx=1n−1tann−1x−∫tann−2xdx∫tannxdx=1n−1tann−1x−∫tann−2xdx$
 Midpoint rule $Mn=∑i=1nf(mi)ΔxMn=∑i=1nf(mi)Δx$ Trapezoidal rule $Tn=12Δx(f(x0)+2f(x1)+2f(x2)+⋯+2f(xn−1)+f(xn))Tn=12Δx(f(x0)+2f(x1)+2f(x2)+⋯+2f(xn−1)+f(xn))$ Simpson’s rule $Sn=Δx3(f(x0)+4f(x1)+2f(x2)+4f(x3)+2f(x4)+4f(x5)+⋯+2f(xn−2)+4f(xn−1)+f(xn))Sn=Δx3(f(x0)+4f(x1)+2f(x2)+4f(x3)+2f(x4)+4f(x5)+⋯+2f(xn−2)+4f(xn−1)+f(xn))$ Error bound for midpoint rule $Error inMn≤M(b−a)324n2Error inMn≤M(b−a)324n2$ Error bound for trapezoidal rule $Error inTn≤M(b−a)312n2Error inTn≤M(b−a)312n2$ Error bound for Simpson’s rule $Error inSn≤M(b−a)5180n4Error inSn≤M(b−a)5180n4$
 Improper integrals $∫a+∞f(x)dx=limt→+∞∫atf(x)dx∫−∞bf(x)dx=limt→−∞∫tbf(x)dx∫−∞+∞f(x)dx=∫−∞0f(x)dx+∫0+∞f(x)dx∫a+∞f(x)dx=limt→+∞∫atf(x)dx∫−∞bf(x)dx=limt→−∞∫tbf(x)dx∫−∞+∞f(x)dx=∫−∞0f(x)dx+∫0+∞f(x)dx$
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