Further Calculus

Question 1

What is an improper integral?

Answer 1

Improper if one or both limits are infinity or if undefined at any point of interval.

Question 2

How to solve improper integral

Answer 2

Use limits like lim t tends to infinity, integral between 0 and t e^-x dx

Question 3

Mean of a function

Answer 3

1/(b-a) integral between a and b f(x)

Question 4

Mean of f(x) + k

Answer 4

Mean of f + k

Question 5

Mean of kf(x)

Answer 5

k x Mean of f

Question 6

cosechx differentiation

Answer 6

-cosechxcothx

Question 7

sechx differentiation

Answer 7

-sechxtanhx

Question 8

cothx differentiation

Answer 8

-cosech^2x

Question 9

Trick for further integration

Answer 9

Replace x^2 to (x+k)^2 if needed

Question 10

How to find sensible/appropriate substitutions

Answer 10

Use formula book as guide to choose between asinu, asinhu, acosu, acoshu. Match to equation

Question 11

Partial fractions tip

Answer 11

When non linear fractions, split and add ax+b, dx+e on top instead of just c or d etc

Question 12

How to do first order differential equations?

Answer 12

Separate the variables
Find integrating factor
Reverse the product rule

Question 13

How to find the integrating factor?

Answer 13

Must be in dy/dx + P(x)y = Q(x) form. Integrating factor is e^int(p)dx

Multiply equation through by IF

Question 14

Any SODE of the form a f’‘(x) + b f’(x) + cx = 0…

Answer 14

has solutions Ae^mx where am^2 + bm + c = 0

Question 15

2 real roots, alpha and beta

Answer 15

Ae^alphax + Be^betax

Question 16

1 real repeated root, alpha

Answer 16

(A+Bx)e^alphax

Question 17

2 imaginary roots, +-iq

Answer 17

Acosqx+Bsinqx

Question 18

2 complex roots, p+-iq

Answer 18

e^px (Acosqx + Bsinqx)

Question 19

a, ax+b, ax^2 + bx + c - Particular Integrals

Answer 19

Same as in question, replace with new letters.

Question 20

ke^px Particular Integral

Answer 20

lambda e^px

Question 21

kxe^px Particular Integral

Answer 21

lambda xe^px + mu e^px

Question 22

mcosqx + nsinqx Particular Integral

Answer 22

lambda cosqx + mu sinqx