Integration

Question 1

how to integrate

Answer 1

increase the power by 1 and divide by the new power (add c)

Question 2

what are integrals with a constant of integration called?

Answer 2

indefinite integrals

Question 3

dividing by 9/4 is the same as…

Answer 3

multiplying by 4/9

Question 4

how to layout first line

Answer 4

∫ (what you are integrating) dx

Question 5

how to check integration

Answer 5

differentiate answer and should get back to question

Question 6

what does dp mean?

Answer 6

to differentiate/integrate with respect to p

Question 7

integrating an expression with multiple terms

Answer 7

integrate each term seperately

Question 8

what should you do before intergrating?

Answer 8

• all brackets should be multiplied out

- there should be no fractions with an x term on the denominator

Question 9

what is a differential equation?

Answer 9

an equation involving derivatives

Question 10

in general, integrating will result in a…

Answer 10

general solution since we can choose the value of c

Question 11

if we have additional information about the function (eg: point it passes through), we can…

Answer 11

find the value of c and obtain a particular solution

Question 12

how to express a function in terms of x

Answer 12

1. integrate
2. sub in x and y values from question to find c
3. write out expression in terms of x

Question 13

what is a definite integral?

Answer 13

when there is no constant of integration and a numerical value is being calculated

Question 14

how to solve definite integrals?

Answer 14

1. integrate as normal
2. evaluate the upper and lower limits
3. upper limit - lower limit

Question 15

in general, the area enclosed by the graph y=f(x) and the x-axis, between x=a and x=b, is given by…

Answer 15

∫ b/a f(x) dx

Question 16

if a negative answer is calculated from a definite integral, what does this indicate?

Answer 16

the area is below the x-axis

can’t have a negative area

Question 17

how to work out total area above and below a graph

Answer 17

work out areas separately then add them together

Question 18

the area between two curves between x=a and x=b is calculated as:

Answer 18

∫ b/a (upper curve - lower curve) dx square units

Question 19

when dealing with areas between curves, areas above and below the x-axis…

Answer 19

do not need to be calculated separately

unless multiple different area

Question 20

how to work out areas between curve and line if no coords are given?

Answer 20

1. equate curve and line to find x coords

2. set up integral and integrate

Question 21

if necessary, what is another way you could integrate to find area?

Answer 21

integrate along the y-axis

Question 22

∫ cos x dx =

Answer 22

sin x + c

Question 23

∫ sin x dx =

Answer 23

-cos x + c

Question 24

what must be used when integrating or differentiating trigonometric function?

Answer 24

radians

Question 25

∫ (ax+b)^n dx =

Answer 25

(ax+b)^n+1 / a(n+1) +c

raise power by 1, divide by new power and also divide by the derivative of the bracket

Question 26

∫ cos (ax+b) dx =

Answer 26

1/a sin (ax+b) + c

Question 27

∫ sin (ax+b) dx =

Answer 27

-1/a cos (ax+b) + c