0195136047.pdf

840 APPENDIX D

D.3 DERIVATIVES AND INTEGRALS

Derivatives:

d

dx

(a)= 0 ,whereais a fixed real number

d

dx

(x)= 1

d

dx

(au)=a

du

dx

,whereuis a function ofx

d

dx

(u±v)=

du

dx

±

dv

dx

,whereuandvare functions ofx

d

dx

(uv)=u

dv

dx

+v

du

dx

d

dx

(

u

v

)=

vdudx−udvdx

v^2

=

1

v

du

dx

−

u

v^2

dv

dx

d

dx

(un)=nun−^1

du

dx

d

dx

[f (u)]=

d

du

[f (u)]·

du

dx

d

dx

(lnu)=

1

u

du

dx

;

d

dx

(lnx)=

1

x

d

dx

(loga u)=(logae)

1

u

du

dx

d

dx

(eu)=eu

du

dx

;

d

dx

eax=aeax

d

dx

(sinu)=

du

dx

(cosu);

d

dx

sinax=acosax

d

dx

(cosu)=−

du

dx

(sinu);

d

dx

cosax=−asinax

Integrals:

∫

(a+bx)ndx=

(a+bx)n+^1

(n+ 1 )b

,n=− 1

∫

dx

a+bx

=

1

b

ln|a+bx|

∫

dx

a^2 +b^2 x^2

=

1

ab

tan−^1

bx

∫ a

xdx

a^2 +x^2

=

1

2

ln(a^2 +x^2 )

∫

x^2 dx

a^2 +x^2

=x−atan−^1

x

∫ a

dx

(a^2 +x^2 )^2

=

x

2 a^2 (a^2 +x^2 )

+

1

2 a^3

tan−^1

x

∫ a

xdx

(a^2 +x^2 )^2

=

− 1

2 (a^2 +x^2 )