Formula Rio Prim Imed

Primitivas Imediatas - Formul´ ario 2008/2009 Sejam f : I −→ R uma fun¸c˜ ao deriv´ avel no intervalo real I e C uma constante real arbitr´ aria.

3.P

f 0 (x) = log |f (x)| + C f (x)

5.P f 0 (x) cos f (x) = sen(f (x)) + C 7.P

f 0 (x) = tg(f (x)) + C cos2 (f (x))

9.P f 0 (x) tg f (x) = − log |cos(f (x))| + C 11.P

¯ ¯ ¯ ¯ f 0 (x) 1 = log ¯¯ + tg f ¯¯ + C cos(f (x)) cos(f (x)) f 0 (x)

13.P p

1 − f 2 (x)

15.P

= arcsen(f (x)) + C

f 0 (x) = arctg(f (x)) + C 1 + f 2 (x)

17.P f 0 (x) ch(f (x)) = sh(f (x)) + C 19.P

f 0 (x) = th(f (x)) + C ch2 (f (x)) f 0 (x)

21.P p

f 2 (x) + 1

23.P

f α+1 (x) + C (α 6= −1) α+1

2.P f 0 (x)f α (x) =

1.P a = ax + C

= argsh(f (x)) + C

f 0 (x) = argth(f (x)) + C 1 − f 2 (x)

4.P af (x) f 0 (x) =

af (x) + C (a ∈ R+ \1) log a

6.P f 0 (x) sen f (x) = − cos f (x) + C f 0 (x) = − cotg(f (x)) + C sen2 f (x)

8.P

10.P f 0 (x) cotg f (x) = log |sen(f (x))| + C 12.P

¯ ¯ ¯ ¯ f 0 (x) 1 = log ¯¯ − cotg(f (x))¯¯ + C sen(f (x)) sen(f (x)) −f 0 (x)

14.P p

1 − f 2 (x)

16.P

= arccos(f (x)) + C

−f 0 (x) = arccotg(f (x)) + C 1 + f 2 (x)

18.P f 0 (x) sh(f (x)) = ch(f (x)) + C 20.P

f 0 (x) = − coth(f (x)) + C sh2 (f (x)) f 0 (x)

22.P p

f 2 (x) − 1

24.P

= argch(f (x)) + C

f 0 (x) = argcoth(f (x)) + C 1 − f 2 (x)

Algumas express˜ oes trigonom´etricas ´ uteis 1. sin(a ± b) = sin a cos b ± cos a sin b 3. cos2 a =

1 + cos(2a) 2

2. cos(a ± b) = cos a cos b ∓ sin a sin b 4. sin2 a =

1 − cos(2a) 2