Universidad De La Salle
Escuela de ciencias b´asicas
Hoja de f´ormulas
1. Hechos ´utiles
1. (a±b)2=a2±2ab +b2
2. a2−b2= (a−b)(a+b)
3. (a+b)3=a3+ 3a2b+ 3ab2+b3
4. (a−b)3=a3−3a2b+ 3ab2−b3
5. a3+b3= (a+b)(a2−ab +b2)
6. a3−b3= (a−b)(a2+ab +b2)
2. Trigonometr´ıa
2.1. Identidades b´asicas
1. sin2x+ cos2x= 1
2. tan2x+ 1 = sec2x
3. cot2x+ 1 = csc2x
4. sin(x±y) = sin xcos y±sin ycos x
5. cos(x±y) = sin xcos y±sin ycos x
6. tan(x±y) = tan x±tan y
1∓tan xtan y
7. sin(2x) = 2 sin xcos x
8. cos(2x) = cos2x−sin2x
9. cos(2x) = 2 cos2x−1=1−2 sin2x
10. sin2x=1−cos(2x)
2
11. cos2x=1+cos(2x)
2
3. Derivadas
3.1. Reglas b´asicas
1. (k)′= 0
2. (xn)′=nxn−1
3. (kf (x))′=kf′(x)
4. (f(x)±g(x))′=f(x)′±g(x)′
5. (f(x)g(x))′=f′(x)g(x) + f(x)g′(x)
6. f(x)
g(x)′=f′(x)g(x)−f(x)g′(x)
(g(x))2
3.2. Regla de la cadena
(f(g(x)))′=f′(g(x))g′(x)
3.3. Funciones trigonom´etricas
1. d
dx sin x= cos x
2. d
dx cos x=−sin x
3. d
dx tan x= sec2x
4. d
dx csc x=−csc xcot x
5. d
dx sec x= sec xtan x
6. d
dx cot x=−csc2x
7. d
dx sin−1x=1
√1−x2
8. d
dx cos−1x=−1
√1−x2
9. d
dx tan−1x=1
1+x2
10. d
dx csc−1x=−1
|x|√x2−1
11. d
dx sec−1x=1
|x|√x2−1
12. d
dx cot−1x=−1
1+x2
3.4. Exponenciales y logaritmos
1. d
dx ln x=1
x
2. d
dx ex=ex
3. d
dx ax=axln a
4. Integrales
1. Rudv =uv −Rvdu
2. Rund=1
n+1 un+1 +C, n =−1
3. R1
udu = ln |u|+C
4. Reudu =eu+C
5. Raudu =au
ln a+C
6. Rsin udu =−cos u+C
7. Rcos udu = sin u+C
8. Rsec2udu = tan u+C
9. Rcsc2udu =−cot u+C
10. Rsec utan udu = sec u+C
11. Rcsc ucot udu =−csc u+C
12. Rtan udu =−ln |cos u|+C
13. Rcot udu =−ln |sin u|+C
14. Rsec udu = ln |sec u+ tan u|+C
15. Rcsc udu = ln |csc u+ cot u|+C
