Home page for accesible maths 1 1 Further Integration

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1.38 Substitutions

factor  substitutionderivative  simplification{\hbox{factor}}\quad\quad{\hbox{substitution}}\quad{\hbox{derivative}}\qquad{% \hbox{simplification}}
a2-x2  x=asinu  dxdu=acosua2-x2=a2cos2u\sqrt{a^{2}-x^{2}}\quad\quad x=a\sin u\qquad{{dx}\over{du}}=a\cos u\quad a^{2}% -x^{2}=a^{2}\cos^{2}u
a2+x2  x=atanu  dxdu=asec2ua2+x2=a2sec2u\sqrt{a^{2}+x^{2}}\quad\quad x=a\tan u\qquad{{dx}\over{du}}=a\sec^{2}u\quad a^% {2}+x^{2}=a^{2}\sec^{2}u
x2-a2  x=acoshudxdu=asinhux2-a2=a2sinh2u.\sqrt{x^{2}-a^{2}}\quad\quad x=a\cosh u\quad{{dx}\over{du}}=a\sinh u\quad x^{2% }-a^{2}=a^{2}\sinh^{2}u.