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Maths Pure Year 2 > Integration > Flashcards

Integration Flashcards

1
Q

E^x

A

E^x

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2
Q

1/x

A

Lnx

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3
Q

Sin x

A

-cos x

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4
Q

Cos x

A

Sin x

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5
Q

Sec^2x

A

Tan x

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6
Q

Cosecx Cotx

A

Cosecx

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7
Q

Cosec^2x

A

-cotx

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8
Q

Secxtanx

A

Secx

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9
Q

E^f(x)

A

1/f’(x) e^f(x)

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10
Q

Cosf(x)

A

1/f’(x) sinf(x)

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11
Q

Sin f(x)

A

-1/f’(x) cos f(x)

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12
Q

Sec^2f(x)

A

1/f’(x) tan f(x)

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13
Q

Cosecf(x)cotf(x)

A

1/ f’(x) cosec f(x)

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14
Q

Cosec^2f(x)

A

-1/f’(x) cotf(x)

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15
Q

Secf(x)tanf(x)

A

1/f’(x) secf(x)

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16
Q

1/f(x)

A

1/f’(x) lnf(x)

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17
Q

1/(f(x))^n when n can’t equal 1

A

1/f’(x)(n+1) F(x)^-(n+1)

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18
Q

Tan x

A

Ln |secx|

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19
Q

Cot x

A

Ln |sinx|

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20
Q

Sec x

A

Ln[ secx +tan x] + c

21
Q

Cosecx

A

-ln [ cosecx + cotx ] + c

22
Q

Ln x

A

X ln x - x + c

23
Q

A^x

A

A^x/ln a

24
Q

How do you differentiate using u sub

A 
Let u equal something in the integral 
Differentiate with respect to x 
Make dx the subject 
Substitute back in 
Something will cancel out 
Swap u back for your blue of x
25
Q

What must you remember when integrating

A

+c

26
Q

Integrating using partial fractions

A

Split into partial fractions

Then differentiate as a function of 1/f(x)

27
Q

What happens if your numerator is bigger than the denominator

A

Must divide first

28
Q

What do you do if you have a definite integra

A

U sub
Then integrate normally as if it is a u sub
Must change the limits to u values rather than x values
Then solve

29
Q

Trig integration with powers of 2 for tan/cot/cosec/sec

A

Use tan trig identity then integrate

30
Q

Trig integration with powers of 2 for sin and cos

A

Use the cos double angle formula

Rearrange then substitute

31
Q

What happens if you have a power of 2 for sin and cos but with f(x) not just x

A

Then alter the double angle formula
E.g. the sin angle was 3x.
Then cos6x= 1-2sin^23x

32
Q

Even powers of sin and cos

A

Factorise so you have your power of 2 inside the brackets then raise it to a higher power
Replace the trig^2 with the double angle formula
Expand the brackets
Solve

33
Q

Odd powers of sin and cos

A

Take a factor of trig^2 out
Use the sin^2 x + cos^2 x = 1
Then let u equal whichever is squared
Normal u sub

34
Q

How to integrate when you have both sin and cos multiplied with different values of x

A

Use the identities sin a+- b ad cos +- b

Re write using the values of x that you have been given in the question
Add or subtract your two values depending on what is in the equation
Create an expression for what is given in the original equation
Should have a basic trig addition or subtraction now to integrate

35
Q

Use of trig identities with no substitution

A

If there is a 1/ square number- x^2 could be rooted or not
Use the trig identities
If there is a root then use s^2 + c^2 = 1
If there isn’t a root then use t^2 + 1 = sec^2

36
Q

How to integrate by parts

A

1st x integral of the second - integrate the (integral of the 2nd x differential of the first )

37
Q

How do you integrate parametric

A

Integral with d and c as limits then y x dx/dt dt

38
Q

What is the trapezium rule

A

Half x width of trapeziums x (ends = (2x middles) )

39
Q

Differential equations by direct integration

A

KEEP DY WHERE IT IS
Separate your variables
Integrate oth sides
Add a plus c on one side

40
Q

If there is a ln in your differential equation what do you do

A

Make the constant in the form of ln

41
Q

If it is a general differentiation then what do you do

A

Leave the c

42
Q

If it is a particular integral the what do you do

A

Need to find c

43
Q

Gradient

A

Dy/dx

44
Q

Acceleration

A

Dv/dt

45
Q

Rate

A

D_/dt

46
Q

Loss

A

Negative

47
Q

Proportional

A

K

48
Q

Inversely proportional

A

K/ _

49
Q

How do you work out a differential equation in context

A

Want = got x need

Maths Pure Year 2 (12 decks)

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