Integrals

Question 1

Indefinite Integrals

Answer 1

Collection of all antiderivative of f with respect to x. When solving, ALWAYS add the constant C.

Question 2

Differential equations

Answer 2

Equivalent to solving dy/dx = f(x), can find the constant C with an initial value of y.

Question 3

x^n dx

Answer 3

x^(n+1)/n+1 + C, n cannot = -1

Question 4

1/x dx

Answer 4

ln|x| + C

Question 5

sinxdx

Answer 5

-cosx + C

Question 6

cosxdx

Answer 6

sinx + C

Question 7

e^x dx

Answer 7

e^x + C

Question 8

tanxdx

Answer 8

ln|secx| + C

Question 9

cotxdx

Answer 9

ln|sinx| + C

Question 10

When can you must use long division to solve an integral?

Answer 10

When the degree of the polynomial in the numerator is greater than or equal to the degree of the divisor (denominator).

Question 11

Long division equation

Answer 11

_______quotient
divisor|numerator
|
remainder

–> numerator/divisor = quotient + remainder/divisor

Question 12

According to the logarithim defined as an integral, what is ln(x)?

Answer 12

integral 1 to x of (1/t)dt

Question 13

What is the growth and decay model?

Answer 13

It is the simplest population model that explains how a population changes over time based on an initial value. The rate of change of population P(t) (function of time) is directly proportional to P(t) for any time ‘t’.

Question 13

Separable differential equations

Answer 13

dy/dx = g(x)*h(y) –> int. 1/h(y)dy = int. g(x)dx + C

Question 14

Growth and Decay model equation

Answer 14

Since dP/dt = kP(t) where K = constant of proportionality, we can rewrite this as a seperable differential equation and integrate to get P(t) = e^(kt)Pinital

Question 15

How does the constant of proportionality (k) act in the growth and decay model?

Answer 15

When K > 0 = growth and K < 0 = decay. Large k = fast change and k = 0 means no change.

Question 16

What is newton’s law of heating/cooling?

Answer 16

The rate of change of temp (dT/dt) of an object is proportional to the difference of temperature of an object and the ambient (surrounding temperature).

Question 17

Newton’s Law of Heating/Cooling Equation

Answer 17

dT/dt = k(Te - T(t)) where Te = fixed temp (ambient) and T(t) = temperature of object. After turning into seperable differential equation and integrating, we have T(t) = Te - (Te - Tnull)e^-(kt)

Question 18

Integration by parts

Answer 18

int. f(x)g(x)dx = int. udv = uv - int. v*du

Question 19

Order for choosing ‘u’ for integration by parts

Answer 19

LIATE - Logs, inverse trig, algebraic, trig, exponents (Some exceptions apply)

Question 20

Trigonometric integrals when at least one power of sine or cosine is odd

Answer 20

Set u = other trig function (the even one even if the other function is absent) and use pythagorean identities

Question 21

Trigonometric integrals when none of the powers of sine or cosine are odd

Answer 21

Use half-angle identities

Question 22

Trigonometric integrals with secant and tangent when the power of tangent is odd

Answer 22

Set u = secx and du = secxtanxdx and use pythagorean identities.

Question 23

Trigonometric integrals with secant and tangent when both tangent and secant have exponents

Answer 23

Set u = tanx and du = sec^2(x)dx and use pythagorean identities.

Question 24

Trigonometric integrals with secant and tangent when tangent is even and secant is odd or is not present

Answer 24

Change tan^2(x) using pythangorean identities

Question 25

Trigonometric integrals involving products of sine and cosine

Answer 25

Use sin(A)sin(B), cos(A)cos(B), sin(A)cos(B) formulas

Question 26

Trig sub a^2 + x^2

Answer 26

x = atan(theta)

Question 27

Trig sub x^2 - a^2

Answer 27

x = asec(theta)

Question 28

Trig sub a^2 - x^2

Answer 28

x = asin(theta)

Question 29

How to integrate rational functions

Answer 29

Fully factor denominator, set fraction equal to broken up version and use algebra to solve for A, B, C etc. Sub in numbers for A, B, C etc and solve broken up integrals - may need to use long division.

Question 30

Rewriting rational functions

Answer 30

1/(linear factor)^n(quadratic factor)^n = (A/linear factor) + (Bx + C)/quadratic, repeat linear and quadratic factors as needed until equal degree of n.

Question 31

What happens when the limit of a definite integral is finite?

Answer 31

The integral converges.

Question 32

What happens when the limit of a definite integral is infinite?

Answer 32

The integral diverges.

Question 33

For what values of p does the int. 1 -> INF 1/x^p converges or diverges?

Answer 33

Converges: p>1 Diverges: p </= 1

Question 34

Direct comparison test

Answer 34

Let f and g be continuous on the interval [a, INF) with 0</= f(x) </= g(x) for all x>/= a. Then:
if int. a-> INF g(x)dx converges, then int. a-> INF f(x)dx also converges.
if int. a-> INF f(x)dx diverges, then int. a-> INF g(x)dx diverges.
Opposite of these is not true.

Question 35

Limit comparison test

Answer 35

If f and g are positive and continuous functions on the interval [a, INF) and if lim(x->INF) f(x)/g(x) = L, 0<L<INF, then:
int. a->INF f(x)dx and int. a->INF g(x)dx either both converge or diverge.

Question 36

For what values of p does the int. 0->1 1/x^p converges or diverges?

Answer 36

converges p < 1, diverges p>/= 1