Flashcards in Methods in Calculus Deck (19)
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1
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)
2
Convergent vs divergent
When the infinite values substituted in make a very small number it converges to the rest of the value produced, else it diverges
3
Integral between a and infinity
Replace infinity with t and use lim t→ ∞ before each line, remove at the end
4
Integral where one value is undefined
Use the limit tending towards the undefined value and integrate either side of it where required
5
Integrate between -∞ and ∞
Integrate from -∞ to 0 and 0 to ∞
6
ȳ for an integral between a and b
a
1/b-a (∫ f(x) dx)
b
7
ȳ of (f(x) + k)
(ȳ of f(x)) + k
8
ȳ of (kf(x))
k(ȳ of f(x))
9
ȳ of (-f(x))
-(ȳ of f(x))
10
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
11
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
12
Proving the integration of inverse trig
Use a substitution to produce an identity
13
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
14
Where the x^2 term in the denominator has a coefficient
Factor it out
15
Two terms in the numerator of integrating inverse trig
Separate them and use reverse chain rule for the term with x in the numerator
16
1/x(x^2+1) partial fractions
A/x + Bx+C/(x^2+1)
17
Larger partial fractions solving method
Equate coefficients and solve that way
18
Partial fractions where the numerator has the same degree as the denominator
Don't use quotient and have an A term with no denominator
19