Y x where f y r f y dy 1 x c a ax c x a x a 1 a 1 c

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( y ( x ) ) where F ( y ) = R f ( y ) dy 1 x + C a ax + C x a x a +1 a +1 + C if a 6 = - 1 1 x ln | x | + C g ( x ) a g 0 ( x ) g ( x ) a +1 a +1 + C if a 6 = - 1 sin x - cos x + C g 0 ( x ) sin g ( x ) - cos g ( x ) + C cos x sin x + C tan x ln | sec x | + C csc x ln | csc x - cot x | + C sec x ln | sec x + tan x | + C cot x ln | sin x | + C sec 2 x tan x + C csc 2 x - cot x + C sec x tan x sec x + C csc x cot x - csc x + C e x e x + C e g ( x ) g 0 ( x ) e g ( x ) + C e ax 1 a e ax + C a x 1 ln a a x + C ln x x ln x - x + C 1 1 - x 2 arcsin x + C g 0 ( x ) 1 - g ( x ) 2 arcsin g ( x ) + C 1 a 2 - x 2 arcsin x a + C 1 1+ x 2 arctan x + C g 0 ( x ) 1+ g ( x ) 2 arctan g ( x ) + C 1 a 2 + x 2 1 a arctan x a + C 1 x 1 - x 2 arcsec x + C
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Properties of Exponentials In the following, x and y are arbitrary real numbers, a and b are arbitrary constants that are strictly bigger than zero and e is 2.7182818284, to ten decimal places. 1) e 0 = 1, a 0 = 1 2) e x + y = e x e y , a x + y = a x a y 3) e - x = 1 e x , a - x = 1 a x 4) ( e x ) y = e xy , ( a x ) y = a xy 5) d dx e x = e x , d dx e g ( x ) = g 0 ( x ) e g ( x ) , d dx a x = (ln a ) a x 6) R e x dx = e x + C , R e ax dx = 1 a e ax + C if a 6 = 0 7) lim x →∞ e x = , lim x →-∞ e x = 0 lim x →∞ a x = , lim x →-∞ a x = 0 if a >
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