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Calculus Volume 2

Key Equations

Calculus Volume 2Key Equations

Key Equations

Integration by parts formula udv=uvvduudv=uvvdu
Integration by parts for definite integrals abudv=uv|ababvduabudv=uv|ababvdu

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((ab)x)12cos((a+b)x)sin(ax)sin(bx)=12cos((ab)x)12cos((a+b)x)
Sine and Cosine Products sin(ax)cos(bx)=12sin((ab)x)+12sin((a+b)x)sin(ax)cos(bx)=12sin((ab)x)+12sin((a+b)x)
Cosine Products cos(ax)cos(bx)=12cos((ab)x)+12cos((a+b)x)cos(ax)cos(bx)=12cos((ab)x)+12cos((a+b)x)
Power Reduction Formula secnx dx=secn-2x tan xn1+n2n1secn2xdx;n1secnx dx=secn-2x tan xn1+n2n1secn2xdx;n1
Power Reduction Formula tannxdx=1n1tann1xtann2xdxtannxdx=1n1tann1xtann2xdx
Midpoint rule Mn=i=1nf(mi)ΔxMn=i=1nf(mi)Δx
Trapezoidal rule Tn=12Δx(f(x0)+2f(x1)+2f(x2)++2f(xn1)+f(xn))Tn=12Δx(f(x0)+2f(x1)+2f(x2)++2f(xn1)+f(xn))
Simpson’s rule Sn=Δx3(f(x0)+4f(x1)+2f(x2)+4f(x3)+2f(x4)+4f(x5)++2f(xn2)+4f(xn1)+f(xn))Sn=Δx3(f(x0)+4f(x1)+2f(x2)+4f(x3)+2f(x4)+4f(x5)++2f(xn2)+4f(xn1)+f(xn))
Error bound for midpoint rule Error inMnM(ba)324n2Error inMnM(ba)324n2
Error bound for trapezoidal rule Error inTnM(ba)312n2Error inTnM(ba)312n2
Error bound for Simpson’s rule Error inSnM(ba)5180n4Error inSnM(ba)5180n4
Improper integrals a+f(x)dx=limt+atf(x)dxbf(x)dx=limttbf(x)dx+f(x)dx=0f(x)dx+0+f(x)dxa+f(x)dx=limt+atf(x)dxbf(x)dx=limttbf(x)dx+f(x)dx=0f(x)dx+0+f(x)dx
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