Trigonometric Relations 1 2 2 2 sin x 1 cos2 x tan x sec x 1 2 1 2 2 2 cos x 1 cos2 x cot x csc x 1 2 1 sin . a c s o b sin a b si n a b 2 1 sin . s n i cos co s a b a b a b 2 1 cos . a c s o b cos a b co s a b 2
Examples 1 1 2 ( ) cos 3 i x dx cos6 x dx 2 1 1 sin6 x x C 2 6 ( ) ii sin 2 x .cos 5 x dx 1 sin 7 x sin 3 x dx 2 1 1 1 cos 7 sin 3 x x C 2 7 3
2 ( ii i ) 1 2sin5 x d x 2 1 4sin5 x 4sin 5 x dx 1 4sin5 x 2 1 cos10 x dx 4 1 x cos5 x 2 x sin10 x C 5 10
2 ( iv ) tan2 x 3sec2 x d x 2 2 tan 2 x 6sec2 x tan2 x 9sec 2 x dx 2 2 sec 2 x 1 6sec2 x tan2 x 9sec 2 x dx 2 6sec2 x tan2 x 10sec 2 x 1 dx 6 10 sec2 tan2 x x x C 2 2
2 2 ( v ) cos 2 . x sin 3 x dx 1 1 1 cos4 x . 1 cos6 x dx 2 2 1 1 cos4 x cos6 x cos4 .cos6 x x dx 4 1 1 1 cos4 x cos6 x cos2 x cos10 x dx 4 2 1 1 1 1 1 1 x sin4 x sin6 x sin2 x sin10 x C 4 4 6 2 2 10
Important Rules n 1 f x n f ' (1) f x x dx C 1 n ' f x ln f x C (2 ) dx f x
Examples 4 1 4 2 x 2 x 2 x 2 x ( ) i e 3 e dx e 3 2 e dx 2 5 2 x e 3 1 C 2 5 1 1/ 2 2 2 3 x 5 6 x dx ( i i ) x 3 x 5 d x 6 3/2 2 3 x 5 1 C 6 3/2
2 2 1 6 x 6 x 1 dx ( i ii ) dx 3 3 6 2 x 6 x 5 2 x 6 x 5 1 ln 2 3 6 5 x x C 6 2 1/ x dx ( ) iv dx 2 ln x ln x x 2lnln x C
1 1 sin x dx sin x dx ( ) v 2 1 x 2 1 x 1 1/ 2 1 sin x dx 2 1 x 3/2 1 sin / 3/2 x C sin3 x 1 3sin3 x dx ( ) vi dx cos3 x 5 3 cos3 x 5 1ln cos3 x 5 C 3
Examples ( ) i tan x dx ( ) ii cot x dx sin x dx - cos x dx - cos x sin x ln cos x - C ln sin x C ( iii ) sec x dx ( iv ) csc x dx sec x tan x csc x cot x csc x dx sec x dx csc x cot x sec x tan x ln csc x cot x C ln sec x tan x C
Integration by Substitution * For many integrals, a substitution can be used to transform the integrand and make possible the finding of an antiderivative. There are a variety of such substitutions, each depending on the form of the integrand. * If a component of the integrand can be viewed as the derivative of another component of the integrand, a substitution can be made to simplify the integrand.
4 5 sin x x dx (1) Example 5 (2) x u 4 5 x dx du (3) Substitute from (2) &(3) into (1) du 4 5 sin x x dx sin u 5 1cos u C 5 1cos 5 x C 5
ln y Example dy (1) 2 y 1 ln y ln y u (2) 1 dy du (3) y Substitute from (2) &(3) into (1) ln y u dy du 2 2 1 u y 1 ln y 1 ln 1 2 u C 2 1 ln 1 2 ln y C 2
2 2tan x Example sec x e dx (1) tan x u (2) 2 (3) sec x dx du Substitute from (2) &(3) into (1) 2 2tan x 2 u sec x e dx e du 1 2 u e C 2 1 2tan x e C 2
sec x tan x dx Example (1) 2 9 4sec x sec x u (2) sec tan x x dx du (3) Substitute from (2) &(3) into (1) sec x tan x dx 1 du 2 2 9 4 u 9 4sec x 3 1 1 a du 2 2 9 4 u 4 1 1 u 1 . tan C 4 3/2 3/2
x Example dx (1) x 1 (2) x u 2 x u dx 2 udu (3) Substitute from (2) &(3) into (1) x u dx 2 udu u 1 x u 1 1 u 2 2 1 u u 2 du - - 2 u 1 u u 1 2 u 1 du u u 1 + + u 1 2 u 1 2 ln 1 u u C 2
1 x dx Example (1) e x e x (2) e u x ln u 1 dx du (3) u Substitute from (2) &(3) into (1) 1 1 1 du x dx e u x 1 u e u 1 du 2 u 1 1 tan u C
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