Integration (P1.13 & 2.11)

Question 1

∫ x^n dx =

Answer 1

x^n+1 / n+1 + c

Question 2

indefinite vs definite integrals

Answer 2

indefinite: + c - produces a function
definite: integral b/w 2 limits - produces a value

Question 3

describe how to find the constant of integration, c

Answer 3

integrate the function
substitute the values (x,y) of a point on the curve into the integrated function
solve the equation to find c

Question 4

describe how to find the definite integral

Answer 4

[f(x)]ba = f(b) - f(a)

Question 5

describe how to find area under curve

Answer 5

definite integral

Question 6

for areas under the x-axis,

Answer 6

integral gives negative answer

Question 7

areas b/w curves & lines

Answer 7

practice

Question 8

∫ 1/x dx =

Answer 8

ln|x| + c

Question 9

∫ e^kx dx =

Answer 9

1/k e^kx + c

Question 10

∫ cosx dx =

Answer 10

sinx + c

Question 11

∫ sinx dx =

Answer 11

-cosx + c

Question 12

describe integration by substitution

Answer 12

substitute a value for u
du/dx =
dx =
cancel out any x’s left
integrate
sub value back into u

Question 13

∫ f’(ax+b) dx =

Answer 13

1/a f(ax+b) + c
reverse chain rule
only works for single powers of x

Question 14

∫ k f’(x)/f(x) dx =

Answer 14

ln|f(x)| & adjust any constant
if integrating a fraction with no trig., look for bottom differentiating to top –> ln

Question 15

tips for integration by parts

Answer 15

∫ uv’ dx = uv - ∫ vu’ dx

u should be a function which becomes simpler after differentiation
it must be possible to integrate v’ to find v
always make lnx term = u
sometimes you need to use integration by parts twice
if vu’ is a product of functions, you have used the wrong u & v (or need to do it twice)

Question 16

practice integration using trig. identities

Question 17

integrate expressions in the form kf’(x)(f(x))^n

Answer 17

f(x))n + 1 & adjust constant

Question 18

practice integration with partial fractions

Question 19

describe Y2 finding areas

Answer 19

the area bounded by 2 curves can be found using integration
R = ∫ba f(x) -g(x) dx
= ∫bc f(x) dx - ∫ba g(x) dx
make sure area under x-axis is made positive before finding total area (to ensure the values for area don’t cancel out)

Question 20

practice exam-style integration questions with transformations

Answer 20

see OneNote

Question 21

tips for trapezium rule

Answer 21

n = # of strips/trapezia
for a convex curve, trapezium rule gives an overestimate for the area
for a concave area, trapezium rule gives an underestimate for the area

Question 22

explain the value for h in the trapezium rule

Answer 22

h = b-a / n
state # of trapezia & each has a height of h
# trapezia x h = b-a

Question 23

explain why trapezium rule is an overestimate of the area under the curve

Answer 23

curve is convex
so lines connecting the 2 endpoints would be above the curve
so it is an overestimate

Question 24

explain why trapezium rule is an underestimate of the area under the curve

Answer 24

curve is concave
so lines connecting the 2 endpoints would be below the curve
so it is an underestimate

Question 25

how is the accuracy of the trapezium rule be improved?

Answer 25

more strips used
decreased width of each strip as more values are used

Question 26

practice trapezium rule

Question 27

describe how to solve differential equations

Answer 27

form differential equation: dy/dx =
separation of variables: put all y terms on the LHS & all x terms & #s on the RHS
see OneNote

Question 28

how do you find the particular solution to a first-order differential equation?

Answer 28

if you know one point on the curve
= boundary condition
practice

Question 29

modelling with differential equations

Question 30

describe integration as the limit of a sum

Answer 30

∫ba f(x) dx = limδ->0 Σb,x=a f(x) δx

you can approximate the area under a curve as a # of thin rectangular strips
as the strips get thinner, the approximation becomes more accurate
limit notation formalises the idea of a definite integral as the limit of a sum of areas of these rectangular strips

how transformations affect area under curve

pg 329 P2 textbook