Home/Calculus II/Appendix B: useful equivalencesSection BAppendix B: useful equivalencesSome antiderivatives can be presented in different forms that differ only by a constant.FormulaCopy LaTeXarsenh (xa)=ln∣x+x2+a2∣−lna,a>0\operatorname{arsenh}\!\left(\frac{x}{a}\right) =\ln\left|x+\sqrt{x^2+a^2}\right|-\ln a, \qquad a>0arsenh(ax)=lnx+x2+a2−lna,a>0a>0a>0a>0View detail →Therefore,FormulaCopy LaTeX∫dxx2+a2=arsenh (xa)+C\int\frac{dx}{\sqrt{x^2+a^2}} =\operatorname{arsenh}\!\left(\frac{x}{a}\right)+C∫x2+a2dx=arsenh(ax)+CView detail →is equivalent to the logarithmic form.Likewise,FormulaCopy LaTeX∫tanx dx=−ln∣cosx∣+C=ln∣secx∣+C.\int\tan x\,dx =-\ln|\cos x|+C =\ln|\sec x|+C.∫tanxdx=−ln∣cosx∣+C=ln∣secx∣+C.View detail →
FormulaCopy LaTeXarsenh (xa)=ln∣x+x2+a2∣−lna,a>0\operatorname{arsenh}\!\left(\frac{x}{a}\right) =\ln\left|x+\sqrt{x^2+a^2}\right|-\ln a, \qquad a>0arsenh(ax)=lnx+x2+a2−lna,a>0a>0a>0a>0View detail →
FormulaCopy LaTeX∫dxx2+a2=arsenh (xa)+C\int\frac{dx}{\sqrt{x^2+a^2}} =\operatorname{arsenh}\!\left(\frac{x}{a}\right)+C∫x2+a2dx=arsenh(ax)+CView detail →
FormulaCopy LaTeX∫tanx dx=−ln∣cosx∣+C=ln∣secx∣+C.\int\tan x\,dx =-\ln|\cos x|+C =\ln|\sec x|+C.∫tanxdx=−ln∣cosx∣+C=ln∣secx∣+C.View detail →