Ambiental

Derivadas
Reglas de derivaci´n o
d [f (x) + g(x)] = f (x) + g (x) dx d [kf (x)] = kf (x) dx d [f (x)g(x)] = f (x)g(x) + f (x)g (x) dx d f (x) f (x)g(x) − f (x)g (x) = dx g(x) g(x)2 d {f [g(x)]} = f [g(x)]g (x) dx d {f (g[h(x)])} = f (g[h(x)])g [h(x)]h (x) dx d (k) = 0 dx d k (x ) = kxk−1 dx Potencia d √ d 1/2 1 ( x) = (x ) = √ dx dx 2 x d dx 1 x = d −1 1 (x ) = − 2 dx x

Suma

ProductoCociente

Regla de la cadena

d [f (x)k ] = kf (x)k−1 f (x) dx d [ dx f (x)] = f (x) 2 f (x)

d 1 f (x) =− dx f (x) f (x)2

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Reglas de derivaci´n (continuaci´n) o o
d (sin x) = cos x dx Trigonom´tricas e d (cos x) = − sin x dx d (tan x) = 1 + tan2 x dx 1 d (arcsin x) = √ dx 1 − x2 Funciones de arco d −1 (arc cos x) = √ dx 1 − x2 d 1 (arctan x) = dx 1 + x2 d x (e ) = ex dx d x (a ) =ax ln a dx d 1 (ln x) = dx x Logar´ ıtmicas d 1 1 (lg x) = dx a x ln a d f (x) 1 (lg f (x)) = dx a f (x) ln a d [sin f (x)] = cos f (x)f (x) dx d [cos f (x)] = − sin f (x)f (x) dx d [tan f (x)] = [1 + tan2 f (x)]f (x) dx d [arcsin f (x)] = dx d [arc cos f (x)] = dx f (x) 1 − f (x)2 −f (x) 1 − f (x)2

d f (x) [arctan f (x)] = dx 1 + f (x)2 d f (x) (e ) = ef (x) f (x) dx d f (x) (a ) = af (x) ln af(x) dx d f (x) (ln f (x)) = dx f (x)

Exponenciales

Integrales
Tabla de integrales inmediatas
xp+1 +C p+1 f (x)p+1 +C p+1

xp dx =

(p = −1)

f (x)p f (x)dx =

(p = −1)

1 dx = ln |x| + C x

f (x) dx = ln |f (x)| + C f (x)

sin xdx = − cos x + C

f (x) sin f (x)dx = − cos f (x) + C

cos xdx = sin x + C

f (x) cos f (x)dx = sin f (x) + C

1 dx = tan x + C cos2 x 1 dx= − cot x + C sin2 x 1 dx = arctan x + C 1 + x2 1 dx = arcsin x + C 1 − x2

f (x) dx = tan f (x) + C cos2 f (x) f (x) dx = − cot f (x) + C sin2 f (x) f (x) dx = arctan f (x) + C 1 + f (x)2 f (x) 1 − f (x)2

√

dx = arcsin f (x) + C

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Tabla de integrales inmediatas (continuaci´n) o
−1 dx = arc cos x + C 1 − x2 −f (x) 1 − f (x)2

√

dx = arc cos f (x) + C

ex dx = ex + C

f(x)ef (x) dx = ef (x) + C

ax dx =

ax +C ln a

f (x)af (x) dx =

af (x) +C ln a

Ejercicios de integrales indefinidas
1. Calcular la integral Soluci´n.o x6 + C. 6 x5 dx.

√ 2. Calcular la integral (x + x)dx. √ x2 2x x Soluci´n.o + + C. 2 3 √ x x 3 √ − 3. Calcular la integral 4 x √ 1 √ Soluci´n.- 6 x − x2 x + C. o 10 4. Calcular la integral Soluci´n.o x2 √ dx. x

dx.

2 2√ x x + C. 54 1 + √ + 2 dx. x2 x x

5. Calcular la integral Soluci´n.- − o

8 1 − √ + 2x + C. x x 1 √ dx. 4 x

6. Calcular la integral Soluci´n.o

4√ 3 4 x + C. 3

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7. Calcular la integral e5x dx. 1 Soluci´n.- e5x + C. o 5 8. Calcular la integral Soluci´n.o cos 5xdx. sin 5x + C. 5

9. Calcular la integral sin axdx. cos ax + C. Soluci´n.- − o a 10. Calcular la integral Soluci´n.o ln x dx. x1 2 ln x + C. 2 1 dx. sin2 3x

11. Calcular la integral Soluci´n.- − o

cot 3x + C. 3 1 dx. cos2 7x

12. Calcular la integral Soluci´n.o

tan 7x + C. 7 1 dx. 3x − 7

13. Calcular la integral Soluci´n.o

1 ln |3x − 7| + C. 3

14. Calcular la integral

1 dx. 1−x Soluci´n.- − ln |1 − x| + C. o 1 dx. 5 − 2x

15. Calcular la integral

1 Soluci´n.- − ln |5 − 2x| + C. o 2 16.Calcular la integral tan 2xdx. 1 Soluci´n.- − ln | cos 2x| + C. o 2 17. Calcular la integral Soluci´n.o sin2 x cos xdx. sin3 x + C. 3 cos3 x sin xdx. cos4 x + C. 4

18. Calcular la integral Soluci´n.- − o

12 √ x x2 + 1dx.

19. Calcular la integral Soluci´n.o 1 3

(x2 + 1)3 + C. √ x dx. 2x2 + 3

20. Calcular la integral Soluci´n.o 1 2

2x2 + 3 + C. cos x dx. sin2 x

21. Calcular laintegral Soluci´n.- − o

1 + C. sin x sin x dx. cos3 x

22. Calcular la integral Soluci´n.o

1 + C. 2 cos2 x tan x dx. cos2 x

23. Calcular la integral Soluci´n.o

tan2 x + C. 2 cot x dx. sin2 x

24. Calcular la integral Soluci´n.- − o

cot2 x + C. 2 ln(x + 1) dx. x+1

25. Calcular la integral Soluci´n.o

ln2 (x + 1) + C. 2 cos x 26. Calcular la integral √ dx. 2 sin x + 1 √...