Page:Calculus Made Easy.pdf/273

${\frac {dx}{dy}}$	← $y$ →	$\int ydx$
Hyperbolic.
$\cosh x$	$\sinh x$	$\cosh x+C$
$\sinh x$	$\cosh x$	$\sinh x+C$
$\operatorname {sech} ^{2}x$	$\tanh x$	$\log _{\epsilon }\cosh x+C$
Miscellaneous.
$-{\frac {1}{(x+a)^{2}}}$	${\frac {1}{x+a}}$	$\log _{\epsilon }(x+a)+C$
$-{\frac {x}{(a^{2}+x^{2})^{\tfrac {3}{2}}}}$	${\frac {1}{\sqrt {a^{2}+x^{2}}}}$	$\log _{\epsilon }(x+{\sqrt {a^{2}+x^{2}}})+C$
$\mp {\frac {b}{(a\pm bx)^{2}}}$	${\frac {1}{a\pm bx}}$	$\pm {\frac {1}{b}}\log _{\epsilon }(a\pm bx)+C$
${\frac {-3a^{x}x}{(a^{2}+x^{2})^{\tfrac {5}{2}}}}$	${\frac {a^{2}}{(a^{2}+x^{2})^{\tfrac {3}{2}}}}$	${\frac {x}{\sqrt {a^{2}+x^{2}}}}+C$
$a\cdot \cos ax$	$\sin ax$	${\frac {1}{a}}\cos ax+C$
$-a\cdot \sin ax$	$\cos ax$	${\frac {1}{a}}\sin ax+C$
$a\cdot \sec ^{2}ax$	$\tan ax$	$-{\frac {1}{a}}\log _{\epsilon }\cos ax+C$
$\sin 2x$	$\sin ^{2}x$	${\frac {x}{2}}-{\frac {\sin 2x}{4}}+C$
$-\sin 2x$	$\cos ^{2}x$	${\frac {x}{2}}+{\frac {\sin 2x}{4}}+C$
$n\cdot \sin ^{n-1}x\cdot \cos x$	$\sin ^{n}x$	${\frac {\cos x}{n}}\sin ^{n-1}x+{\frac {n-1}{n}}\int \sin ^{n-2}xdx+C$
$-{\frac {\cos x}{\sin ^{2}x}}$	${\frac {1}{\sin x}}$	$\log _{\epsilon }\tan {\frac {x}{2}}+C$
$-{\frac {\sin 2x}{\sin ^{4}x}}$	${\frac {1}{\sin ^{2}x}}$	$-\operatorname {cotan} x+C$
${\frac {\sin ^{2}x-\cos ^{2}x}{\sin ^{2}x\cdot \cos ^{2}x}}$	${\frac {1}{\sin x\cdot \cos x}}$	$\log _{\epsilon }\tan x+C$
$n\cdot \sin mx\cdot \cos nx+m\cdot \sin nx\cdot \cos mx$	$\sin mx\cdot \sin nx$	${\tfrac {1}{2}}\cos(m-n)x-{\tfrac {1}{2}}\cos(m+n)x+C$
$2a\cdot \sin 2ax$	$\sin ^{2}ax$	${\frac {x}{2}}-{\frac {\sin 2ax}{4a}}+C$
$-2a\cdot \sin 2ax$	$\cos ^{2}ax$	${\frac {x}{2}}+{\frac {\sin 2ax}{4a}}+C$