Y2, C11 - Integration Flashcards
Integral of x^n
(1 / n+1) * x^n+1 + c
Integral of e^x
e^x + c
Integral of 1 / x
ln(x) + c
Integral of cosx
sinx + c
Integral of sinx
-cosx + c
Integral of sec^2(x)
tanx + c
Integral of cosecx * cotx
-cosecx + c
Integral of cosec^2(x)
-cotx + c
Integral of secxtanx
secx + c
Integrate f’(ax + b) dx
1/a f(ax + b) + C
Integrate (10x + 11)^12
(1/130)(10x + 11)^13 + C
Integrate sin(3x)cos(3x)
1/2 sin(6x) = sin3xcos3x
Therefore ans = -1/12 cos6x + C
OR
ans = 1/6 * sin^2(3x) + C
What are the steps of the reverse chain rule
1) Consider some expression that will differentiate to something similar to it
2) Differentiate and then scale for any difference
When integrating, what should you try if the bottom fraction differentiates to give the top fraction
Try ln of the bottom
When differentiating sin or sec with exponents, what happens to the exponent (power)
sin –> power decreases
sec –> power stays the same
What are the steps for integration by substitution
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
What can you do if you have a constant factor within an integral
Take it out to the front
What are sensible substitutions to use
Expressions inside roots, powers, or the denominator of a fraction
What are the steps for definite integration by parts
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)
What is the formula for integration by parts
int(u * dv/dx) dx = uv - int(v * du/dx) dx
OR
int(u * v’) dx = uv - int(v * u’) dx
When would you use integration by parts
When integrating a product
Using the ‘LIATE’ list, choose ‘u’ to be the function which comes first in the list, what is the list
L - logarithmic function
I - inverse trig function
A - algebraic function
T - trig function
E - exponential function
How would you integrate algebraic fractions
Using partial fractions
How would you integrate 2 / (x^2 - 1) dx
Using partial fractions
= lnl (x-1)/(x+1) l + c
How to integrate top-heavy algebraic fractions
Algebraic long division
Then integrate using partial fractions
When working out the area between curves, how do you deal with the integral of f(x) and the integral of g(x)
Combine the integrals when subtracting f(x) from g(x)
= integral from b to a of (f(x) - g(x)) dx
What are possible techniques of integration
Substitution
Int. by parts (ln take priority)
Reverse chain rule
Impartial fractions
Standard results
Use trig identities
Polynomial division
Split the numerator
What does integration by parts usually look like
Integration with products
What technique would you use to integrate ln(x)
Integration by parts
What is the formula for the area of a trapezium
1/2 (a + b)h
When would the trapezium be an overestimate
When f(x) is convex (bends upwards)
When would the trapezium rule be an underestimate
When f(x) is concave (bends downwards
When is f(x) convex
When f’‘(x) > 0
When if f(x) concave
f’‘(x) < 0
What is the formula for integrating using the trapezium rule
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)
What is the formula for percentage error
(change / original) * 100
= ((New - actual) / actual) * 100
What is the equation for parametric integration
int (y) dx =
int(y * (dx/dt)) dt
What happens to the limits of parametric integration
The limits must be changed to be in terms of t
Change the x values to t values
What are the steps for parametric integration
1) Find dx / dt
2) Change limits
3) Sub into formula
4) Integrate
Find the general solution to dy/dx = xy + y
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
What makes a general solution to a parametric equation ‘general’
The unknown constant A or c or k
What should you do when you have lots of ln’s
Combine them into one ln
When integrating differential equations, what should you do to the constant of integration if you have ln on the RHS
Make the constant of integration lnlkl or lnlAl
How do you know when to use the reverse chain rule
When the numerator is the derivative of the denominator
When one factor of a product expression is related to the derivative of the other
When can you split the numerator
When there is a single term in the denominator
When you seen an integration, what is the order of methods you should try?
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)
