Skip to ContentGo to accessibility pageKeyboard shortcuts menu
OpenStax Logo
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
Order a print copy

As an Amazon Associate we earn from qualifying purchases.

Citation/Attribution

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution-NonCommercial-ShareAlike License and you must attribute OpenStax.

Attribution information
  • If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
    Access for free at https://openstax.org/books/calculus-volume-2/pages/1-introduction
  • If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
    Access for free at https://openstax.org/books/calculus-volume-2/pages/1-introduction
Citation information

© Feb 5, 2024 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.