Methods in Calculus Flashcards Preview

A Level Further Maths Pure 2 > Methods in Calculus > Flashcards

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

arctan(∞)

π/2