Integration

Question 1

Generally start with a function f and want to find another function F with derivative f

Answer 1

Such a function F is the antiderivative

Question 2

Indefinite integral

Answer 2

eg F(x) = x^2 + C is indefinite integral of f(x) = 2x as it describes all antiderivatives of f(x)=2x

Question 3

Diff between antiderivative and indefinite integral

Answer 3

Indefinite integral describes all antiderivatives for a function by having the arbitrary constant added to a function. Antiderivative of f is a function whose derivative is f so a specific function whereas indefinite integral describes ALL antiderivatives of f

Question 4

Antiderivatives of power functions

Answer 4

Increase power by 1 then divide by the new power. Eg x^10 becomes x^11, then 1/11*x^11.

Question 5

Constant multiple rule for antiderivatives

Answer 5

F(x) antiderivative of f(x), k is constant. Then kF(x) is an antiderivative of kf(x). Eg 10x^4, second part is 1/5x^5. So 10* 1/5x^5, so 2x^5 (10 * 1/5=2)

Question 6

Sum rule for antiderivatives

Answer 6

if F(x) and G(x) derivatives of f(x) and g(x) then F(x)+G(x) is an antiderivative of f(x)+g(x)

Question 7

Indefinite integral of a constant function

Answer 7

eg Indefinite integral of a is ax+c. eg of 10 is 10x + c

Question 8

Finding a particular antiderivative.

Answer 8

eg f(x)=x^2+5 such that F(3)=20. Indefinite integral is 1/3x^3 + 5x + c. Substitute 3 for x in indefinite integral
=9+15+c
9+15+c=20
c = -4 so 1/3x^3+5x-4 is required antiderivative

Question 9

Changes in the values of antiderivatives

Answer 9

To find how much an antiderivative value changes, find ANY antiderivative for that function, then substitute values you want to see difference for and do one minus the other. Eg will give you change in quantity if you know rate of change (qty is rate of change integrated)

Question 10

Antiderivative of e^(x/8)

Answer 10

8e^(x/8)

Question 11

Antiderivative of reciprocal, eg 1/x, 4/x, 1/2x

Answer 11

eg of 1x it is ln|x|+c. If x only takes pos nos then lnx + C (mod not needed)

Question 12

Area approximation

Answer 12

equal subintervals, left endpoints on each on curve. Left endpoint of sub int x b-a/n. b is right end of interval, a is left end. Basically width of subint x height of sub int (got from putting x value into function). Add them all together to give area, or use formula for signed area

Question 13

Signed area

Answer 13

+ if entirely above x axis, - if below.

Question 14

Signed area formula

Answer 14

f(left endpoint of sub int nearest a)x(b-a/n)). So height by width of each sub interval

Question 15

Signed area, if b larger than a

Answer 15

eg from 5 to 3. Same as 3 to 5 but with reversed sign…so if 3 to 5 signed area is 2, 5 to 3 is -2

Question 16

Definite integral

Answer 16

∫sign, numbers bottom and top f(x) dx (signed area)

Question 17

Algebraic definition of a definite integral

Answer 17

∫sign, numbers bottom and top f(x)dx = lim/n to infinity (f(a+0w)+f(a+1w)+f(a+2w)…)w. w-(b-a)/n

Question 18

Fundamental theorem of calculus

Answer 18

Find signed area when you know a formula for an antiderivative F, of f. ∫sign numbers top and bottom f(x) dx = F(b)-F(a)

Question 19

Fundamental theorem [square bracket notation]

Answer 19

f is continuous function, F is an antiderivative of f…∫sign, numbers bottom and top f(x)dx = [F(x)] numbers bottom and top

Question 20

Finding signed area

Answer 20

Same as evaluating a definite integral

Question 21

Integrand

Answer 21

inside the integration notation…eg x^2 if ∫sign, numbers bottom and top f(x) x^2 dx

Question 22

Constant sum and multiple rule for def integral and square bracket notation

Answer 22

[kF(x)]numbers top and bottom = k[F(x)]numbers top and bottom, where k is constant.
[F(x)+G(x)]numbers top and bottom = [F(x)] nos top and bottom + [G(x)] nos top and bottom. Applies for both square bracket notation and definite integrals

Question 23

Notation for indefinite integral

Answer 23

eg ∫ sec^2 x dx = tan x +c ALWAYS with arbitrary constant. NEVER use with an antiderivative, just indefinite integral

Question 24

rearrange integrand so can integrate

Answer 24

eg ∫(x-5/x) dx….∫1-( 5/x)dx…..∫1 dx - 5∫1/x dx

Question 25

Integration by substitution

Answer 25

Reverse of chain rule. so, something by the derivative of something eg cos x^2 (2x) Something is x^2 so set as u
Then write in format ∫ (u) du…so cos u du….then integrate so sin u + c…sin (x^2) + c. Basically forgetting about the derivative of something when you integrate, so find the something, substitute u for it, ignore its derivative, then rewrite with u, then integrate, then replace u with whatever u stood for

Question 26

When substituting

Answer 26

take constant with u if possible… eg (sin (5+2x^3)) (6x^2).
u=(5+2x^3).

Question 27

Rearranging before substitution

Answer 27

You can multiply by a constant INSIDE the integral and dividing by the same constant OUTSIDE the integral. If, say, te derivative of something is 3x^2 and there is only x^2 in whatever needs integrating, you can count it as the derivative of something then multiply it by 3 inside, and divide by 3 outside. In effect, divide by whatever outside and ignore the inside multiplication (as it involves the derivative of u)

Question 28

If something and somethings derivative in expression to be integrated

Answer 28

u is something, then integrate u

Question 29

If something’s derivative NOT in expression

Answer 29

then can divide and multiply by a constant that would make the derivative. Eg if there is 2x^2 and x (derivative should be 4x) divide by 4 outside of integral

Question 30

Rearrange - when there is for example 9x^2

Answer 30

rearrange to (3x)^2 so can take u=3x

Question 31

indefinite integral of a function of a linear expression

Answer 31

∫ f(ax+b) dx = 1/a F(ax+b) +c (if you know an antiderivative). eg cos (3x-2) dx = 1/3 sin(3x-2) + c

Question 32

indefinite integral of a function of a multiple of the variable

Answer 32

∫f(ax) dx = 1/a F (ax) + c eg sin (1/5 x) dx = -5 cos (1/5 x) + c sin is f, cos is F (the antiderivative), (1/5) is a, x is x

Question 33

Integration by parts

Answer 33

Opposite of product rule, so integrand product of 2 expressions. ∫ f(x)g(x) dx = f(x)G(x) - ∫ f’(x)G(x) dx G is an antiderivative of g. Can use IF you can find an antiderivative of g. eg ∫ x sin x dx (so g is sin x, therefore G=-cos x.