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Summary Table Of Integrals

This page is a draft and is under active development. 

( \newcommand{\kernel}{\mathrm{null}\,}\)

Indefinite Integral

uαdu=uα+1α+1+c,α1duu=ln|u|+ccosudu=sinu+csinudu=cosu+ctanudu=ln|cosu|+ccotudu=ln|sinu|+csec2udu=tanu+ccsc2udu=cotu+csecudu=ln|secu+tanu|+ccsc(u)du=ln|csc(u)cot(u)|+ccos2udu=u2+14sin2u+csin2udu=u214sin2u+cdu1+u2du=tan1u+cdu1u2du=sin1u+c1u21du=12ln|u1u+1|+c

 

Integration Rules

(Af(x)+Bg(x)dx=Af(x)dx+Bg(x)dx

 f(g(x))g(x)dx=f(g(x))+C

Definite Integral

aaf(x)dx=0

abf(x)dx=baf(x)dx

ba[f(x)+g(x)]dx=baf(x)dx+bag(x)dx

ba[f(x)g(x)]dx=baf(x)dxbag(x)dx

bacf(x)dx=cbaf(x)

for constant c. baf(x)dx=caf(x)dx+bcf(x)dx

Although this formula normally applies when c is between a and b, the formula holds for all values of a, b, and c, provided f(x) is integrable on the largest interval.

Let f be continuous on [a,b] and let F be any anti-derivative of f. Then baf(x)dx=F(b)F(a).

 Reduction formulas

sinn(x) dx=1nsinn1xcosx+n1nsinn2xdx.

cosn(x) dx=1ncosn1xsinx+n1ncosn2xdx.

tann(x) dx=1n1tann1xtann2x dx,n1.

secn(x) dx=1n1secn2xtanx+n2n1secn2x dx,n2.

Integration by parts

udv=uvvduucosudu=usinu+cosu+cusinudu=ucosu+sinu+cueudu=ueueu+ceλucosωudu=eλu(λcosωu+ωsinωu)λ2+ω2+ceλusinωudu=eλu(λsinωuωcosωu)λ2+ω2+cln|u|du=uln|u|u+culn|u|du=u2ln|u|2u24+ccosω1ucosω2udu=sin(ω1+ω2)u2(ω1+ω2)+sin(ω1ω2)u2(ω1ω2)+c(ω1±ω2)sinω1usinω2udu=sin(ω1+ω2)u2(ω1+ω2)+sin(ω1ω2)u2(ω1ω2)+c(ω1±ω2)sinω1ucosω2udu=cos(ω1+ω2)u2(ω1+ω2)cos(ω1ω2)u2(ω1ω2)+c(ω1±ω2)

Improper Integrals

Let f(x) be continuous over an interval of the form [a,+). Then +af(x)dx=lim provided this limit exists.

 Let f(x) be continuous over an interval of the form (-\infty,b]. Then \int ^b_{-\infty}f(x)dx=\lim_{t \to -\infty}\int ^b_tf(x)dx, \label{improper2} provided this limit exists.

In each case, if the limit exists, then the improper integral is said to converge. If the limit does not exist, then the improper integral is said to diverge.

Let f(x) be continuous over (-\infty,+\infty). Then \int ^{+\infty}_{-\infty}f(x)dx=\int ^0_{-\infty}f(x)dx+\int ^{+\infty}_0f(x)dx, \label{improper3} provided that \int ^0_{-\infty}f(x)dx and \int ^{+\infty}_0f(x)dx both converge. If either of these two integrals diverge, then \int ^{+\infty}_{-\infty}f(x)dx diverges. (It can be shown that, in fact, \int ^{+\infty}_{-\infty}f(x)dx=\int ^a_{-\infty}f(x)dx+\int ^{+\infty}_af(x)dx for any value of a.).

Let f(x) be continuous over [a,b). Then, \int ^b_af(x)dx=\lim_{t \to b^-}\int ^t_af(x)dx.

Let f(x) be continuous over (a,b]. Then, \int ^b_af(x)dx=\lim_{t \to a^+}\int ^b_tf(x)dx. In each case, if the limit exists, then the improper integral is said to converge. If the limit does not exist, then the improper integral is said to diverge.

 If f(x) is continuous over [a,b] except at a point c in (a,b), then \int ^b_af(x)dx=\int ^c_af(x)dx+\int ^b_cf(x)dx, provided both \int ^c_af(x)dx and \int ^b_cf(x)dx converge. If either of these integrals diverges, then \int ^b_af(x)dx diverges.

 

   

 


This page titled Summary Table Of Integrals is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Pamini Thangarajah.

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