What is an improper integral?
Improper if one or both limits are infinity or if undefined at any point of interval.
How to solve improper integral
Use limits like lim t tends to infinity, integral between 0 and t e^-x dx
Mean of a function
1/(b-a) integral between a and b f(x)
Mean of f(x) + k
Mean of f + k
Mean of kf(x)
k x Mean of f
cosechx differentiation
-cosechxcothx
sechx differentiation
-sechxtanhx
cothx differentiation
-cosech^2x
Trick for further integration
Replace x^2 to (x+k)^2 if needed
How to find sensible/appropriate substitutions
Use formula book as guide to choose between asinu, asinhu, acosu, acoshu. Match to equation
Partial fractions tip
When non linear fractions, split and add ax+b, dx+e on top instead of just c or d etc
How to do first order differential equations?
Separate the variables
Find integrating factor
Reverse the product rule
How to find the integrating factor?
Must be in dy/dx + P(x)y = Q(x) form. Integrating factor is e^int(p)dx
Multiply equation through by IF
Any SODE of the form a f’‘(x) + b f’(x) + cx = 0…
has solutions Ae^mx where am^2 + bm + c = 0
2 real roots, alpha and beta
Ae^alphax + Be^betax
1 real repeated root, alpha
(A+Bx)e^alphax
2 imaginary roots, +-iq
Acosqx+Bsinqx
2 complex roots, p+-iq
e^px (Acosqx + Bsinqx)
a, ax+b, ax^2 + bx + c - Particular Integrals
Same as in question, replace with new letters.
ke^px Particular Integral
lambda e^px
kxe^px Particular Integral
lambda xe^px + mu e^px
mcosqx + nsinqx Particular Integral
lambda cosqx + mu sinqx
