Y2, C11 - Integration

Question 1

Integral of x^n

Answer 1

(1 / n+1) * x^n+1 + c

Question 2

Integral of e^x

Answer 2

e^x + c

Question 3

Integral of 1 / x

Answer 3

ln(x) + c

Question 4

Integral of cosx

Answer 4

sinx + c

Question 5

Integral of sinx

Answer 5

-cosx + c

Question 6

Integral of sec^2(x)

Answer 6

tanx + c

Question 7

Integral of cosecx * cotx

Answer 7

-cosecx + c

Question 8

Integral of cosec^2(x)

Answer 8

-cotx + c

Question 9

Integral of secxtanx

Answer 9

secx + c

Question 10

Integrate f’(ax + b) dx

Answer 10

1/a f(ax + b) + C

Question 11

Integrate (10x + 11)^12

Answer 11

(1/130)(10x + 11)^13 + C

Question 12

Integrate sin(3x)cos(3x)

Answer 12

1/2 sin(6x) = sin3xcos3x
Therefore ans = -1/12 cos6x + C
OR
ans = 1/6 * sin^2(3x) + C

Question 13

What are the steps of the reverse chain rule

Answer 13

1) Consider some expression that will differentiate to something similar to it
2) Differentiate and then scale for any difference

Question 14

When integrating, what should you try if the bottom fraction differentiates to give the top fraction

Answer 14

Try ln of the bottom

Question 15

When differentiating sin or sec with exponents, what happens to the exponent (power)

Answer 15

sin –> power decreases
sec –> power stays the same

Question 16

What are the steps for integration by substitution

Answer 16

1) Using substitution, work out x and dx (or variant)
2) Substitute these into expression
3) Integrate simplified expression
4) Write answer in terms of x

Question 17

What can you do if you have a constant factor within an integral

Answer 17

Take it out to the front

Question 18

What are sensible substitutions to use

Answer 18

Expressions inside roots, powers, or the denominator of a fraction

Question 19

What are the steps for definite integration by parts

Answer 19

1) Using substitution, work out x and dx (or variant)
2) Substitute these into expression
3) Integrate simplified expression
4) Write answer in terms of x
Ex) Change limits to match the ‘dx’ part (make the limits u limits)

Question 20

What is the formula for integration by parts

Answer 20

int(u * dv/dx) dx = uv - int(v * du/dx) dx
OR
int(u * v’) dx = uv - int(v * u’) dx

Question 21

When would you use integration by parts

Answer 21

When integrating a product

Question 22

Using the ‘LIATE’ list, choose ‘u’ to be the function which comes first in the list, what is the list

Answer 22

L - logarithmic function
I - inverse trig function
A - algebraic function
T - trig function
E - exponential function

Question 23

How would you integrate algebraic fractions

Answer 23

Using partial fractions

Question 24

How would you integrate 2 / (x^2 - 1) dx

Answer 24

Using partial fractions
= lnl (x-1)/(x+1) l + c

Question 25

How to integrate top-heavy algebraic fractions

Answer 25

Algebraic long division
Then integrate using partial fractions

Question 26

When working out the area between curves, how do you deal with the integral of f(x) and the integral of g(x)

Answer 26

Combine the integrals when subtracting f(x) from g(x)
= integral from b to a of (f(x) - g(x)) dx

Question 27

What are possible techniques of integration

Answer 27

Substitution
Int. by parts (ln take priority)
Reverse chain rule
Impartial fractions
Standard results
Use trig identities
Polynomial division
Split the numerator

Question 28

What does integration by parts usually look like

Answer 28

Integration with products

Question 29

What technique would you use to integrate ln(x)

Answer 29

Integration by parts

Question 30

What is the formula for the area of a trapezium

Answer 30

1/2 (a + b)h

Question 31

When would the trapezium be an overestimate

Answer 31

When f(x) is convex (bends upwards)

Question 32

When would the trapezium rule be an underestimate

Answer 32

When f(x) is concave (bends downwards

Question 33

When is f(x) convex

Answer 33

When f’‘(x) > 0

Question 34

When if f(x) concave

Answer 34

f’‘(x) < 0

Question 35

What is the formula for integrating using the trapezium rule

Answer 35

integral from b to a of (y) dx =
h/2 (y1 + 2(y2 + … + yn-1) + yn)
Where h = width of each trapezium
The middle trapeziums are double (y2 + … + yn-1)
The end values are only used once (y1 and yn)

Question 36

What is the formula for percentage error

Answer 36

(change / original) * 100
= ((New - actual) / actual) * 100

Question 37

What is the equation for parametric integration

Answer 37

int (y) dx =
int(y * (dx/dt)) dt

Question 38

What happens to the limits of parametric integration

Answer 38

The limits must be changed to be in terms of t
Change the x values to t values

Question 39

What are the steps for parametric integration

Answer 39

1) Find dx / dt
2) Change limits
3) Sub into formula
4) Integrate

Question 40

Find the general solution to dy/dx = xy + y

Answer 40

dy/dx = xy + y
dy/dx = y(x+1)
1/y * dy/dx = x+1
int(1/y) dy = int(x+1) dx
lnlyl = 1/2 * x^2 + x + c
y = e^0.5x^2 + x + c
y = Ae^0.5x^2 + x
Where A = e^c

Question 41

What makes a general solution to a parametric equation ‘general’

Answer 41

The unknown constant A or c or k

Question 42

What should you do when you have lots of ln’s

Answer 42

Combine them into one ln

Question 43

When integrating differential equations, what should you do to the constant of integration if you have ln on the RHS

Answer 43

Make the constant of integration lnlkl or lnlAl

Question 44

How do you know when to use the reverse chain rule

Answer 44

When the numerator is the derivative of the denominator
When one factor of a product expression is related to the derivative of the other

Question 45

When can you split the numerator

Answer 45

When there is a single term in the denominator

Question 46

When you seen an integration, what is the order of methods you should try?

Answer 46

1) Standard result (scaling?)
2) Manipulate to standard result (expand brackets / trig identities)
3) Reverse chain rule (is numerator derivative of denominator, is one factor related to the derivative of the other)
FRACTIONAL EXPRESSIONS
4a) Split numerator (single term in denominator)
4b) Partial fractions (does denominator factorise)
4c) Algebraic division (is fraction improper)
PRODUCT EXPRESSIONS
4) Integration by parts (for u, choose ln term, then polynomial)
5) Substitution (LAST RESORT)