Integration Flashcards
how to integrate
increase the power by 1 and divide by the new power (add c)
what are integrals with a constant of integration called?
indefinite integrals
dividing by 9/4 is the same as…
multiplying by 4/9
how to layout first line
∫ (what you are integrating) dx
how to check integration
differentiate answer and should get back to question
what does dp mean?
to differentiate/integrate with respect to p
integrating an expression with multiple terms
integrate each term seperately
what should you do before intergrating?
- all brackets should be multiplied out
- there should be no fractions with an x term on the denominator
what is a differential equation?
an equation involving derivatives
in general, integrating will result in a…
general solution since we can choose the value of c
if we have additional information about the function (eg: point it passes through), we can…
find the value of c and obtain a particular solution
how to express a function in terms of x
- integrate
- sub in x and y values from question to find c
- write out expression in terms of x
what is a definite integral?
when there is no constant of integration and a numerical value is being calculated
how to solve definite integrals?
- integrate as normal
- evaluate the upper and lower limits
- upper limit - lower limit
in general, the area enclosed by the graph y=f(x) and the x-axis, between x=a and x=b, is given by…
∫ b/a f(x) dx
if a negative answer is calculated from a definite integral, what does this indicate?
the area is below the x-axis
can’t have a negative area
how to work out total area above and below a graph
work out areas separately then add them together
the area between two curves between x=a and x=b is calculated as:
∫ b/a (upper curve - lower curve) dx square units
when dealing with areas between curves, areas above and below the x-axis…
do not need to be calculated separately
unless multiple different area
how to work out areas between curve and line if no coords are given?
- equate curve and line to find x coords
2. set up integral and integrate
if necessary, what is another way you could integrate to find area?
integrate along the y-axis
∫ cos x dx =
sin x + c
∫ sin x dx =
-cos x + c
what must be used when integrating or differentiating trigonometric function?
radians
∫ (ax+b)^n dx =
(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
∫ cos (ax+b) dx =
1/a sin (ax+b) + c
∫ sin (ax+b) dx =
-1/a cos (ax+b) + c
