Methods in Calculus Flashcards
When is an integral improper?
-One or both of the limits is infinite
-f(x) is undefined at x = a,b or any point in the interval [a.b] (generally from dividing by 0)
Convergent vs divergent
Convergence is when the limit as x -> infinity tends to a value. Divergence is where it doesn’t tend to a value.
Integral between a and infinity
Replace infinity with t and use lim t→ ∞ before each line, remove at the end
Integral where one value is undefined
Use the limit tending towards the undefined value and integrate either side of it where required
Integrate between -∞ and ∞
Integrate from -∞ to 0 and 0 to ∞
ȳ for an integral between a and b
a
1/b-a (∫ f(x) dx)
b
ȳ of (f(x) + k)
(ȳ of f(x)) + k
ȳ of (kf(x))
k(ȳ of f(x))
ȳ of (-f(x))
-(ȳ of f(x))
Differentiating inverse trig proof
Use the trig on the other side, take dx/dy, find the reciprocal and use identities to get it in terms of x
When to use proof of inverse trig and when to use formulae
Use proof if it is a show that, else use formulae, remembering full chain rule if it isn’t just x
Proving the integration of inverse trig
Use a substitution to produce an identity
Dealing with 1/a term of a^2 and x^2
Factor out the a^2 and integrate using the 1+- x^2 rules using u = x/a
Where the x^2 term in the denominator has a coefficient
Factor it out
Two terms in the numerator of integrating inverse trig
Separate them and use reverse chain rule for the term with x in the numerator
