U5 - Integrals Part 2 (u sub)
What are frequent choices to pick u as? remember, it is what is inside that would be taken as the inside in the chain rule.
- inside functions - denominators - an exponent
∫csc u du
- ln | cscu + cotu | + C
∫tan u du
-ln | cos u | + C
How do you take u substitution for Definite Integral? What are the 2 methods?
1. Evaluation without a change of limits of integration 2. Evaluation with a change of limits of inegration
How do you take inverse trig integrals?
1. determine which inverse trig format it would match to integrate into inverse trig function - 1/u^2 + 1 = arctan - 1/√1-u^2 = arcsin - 1/u(√u^2 - 1) = arcsec 2. format the integrand to be able to take a u substitution for the x^2 value and simplify strategies: - divide the denominator to make it match the 1+/1- values in the denominator --> move divisor into the x^2 value by dividing/multiplying by the divisor's root - take out u as x^2, or some value, to make the denominator go to a inverse trig value - complete the square in the denominator, then u substitute 3. once in format, just make integral equal to inverse trig function +C, or defined integral, number answer
Workflow for solving integrals with dif methods?
1. does long division work? (degree rule) 2. does completing the square work, and which inverse sin function would it form into? 3. can I take out a simple u value to turn it into an inverse trig function? 4. If I did a u-sub, will du cancel out? If there are extra terms, I should just replace them with their u values.
How to do u substitution of integrals?
1. set u equal to the inside function (inside parenthesis, under root, denom, etc.) 2. solve for du in terms of dx, and plug in du value 3. take integral in terms of u 4. plug in u value for originial x
How do u do u substitution of definite integral with a change to limits of integration?
1. set up normal u substitution with u and du 2. plug in limits of integration into u = x + c (ex) to get limits of integration in terms of u 3. take whole integral and completely solve out without plugging in value for u, since limits of integration changed, you don't have to
how to do implicit differentiation
1. take deriv of both sides with respect to x (all unmatched vars have dv/dx) 2. collect all dy/dx to one side 3. factor out dy/dx and isolate 4. solve for dy/dxdy/dx is the derivative, the rate of change of y with respect to xchange in y/change in x @ infinitly small interval ALL UNMATCHED VARIABLES TO DX GET D(Y)/DX
How do you do u substitution of definite integral without a change to limits of integration?
1. take u as inside function as normal, and du 2. take integral of udu, with limits of integration as x = __ & x= __ 3. take integral in terms of u, with evaluation symbol with limits still as x =__ & x=__ 4. Plug in original function for u, then replace in x limits and do F(x2) - F(x1) to evaluate integral
When do you plug in x for it's value in terms of u for u-sub, instead of using other methods like completing the square, long division, u-sub into inverse trig, etc.
Check for all other methods first - long division completing the square to inverse trig other ways to get to inverse trig formulas If a u sub is possible, with just extra x terms in the way that do not cancel out with du, replace them with their value as u.
What is the reasoning to choose a u value?
Choose a u value so that du is equal to the other variables/expressions in the integrand, in order to cancel out **this is necessary, always choose a u that can cancel out as many variables as possible in integrand
Generally, if there is no power greater/equal to in numerator, and no other terms that can cancel out with u substitution, what type of integration is it?
Completing the square!! - u substitution can get creative, don't rule it out immediately
Workflow for completing the square integration?
Factor the denominator by completing the square - add (b/2)^2, and subtract (b/2)^2 outside () re-arrange integrand till it matches the inverse trig function derivative to simplify (divide, root inside square, factor) take integration into arctrig function - should look like with coefficient 1/x before trig function, and 1/x inside () - general format
How do accumulating rates work? adding one? subtracting one?
If R(x) is the rate at which sand arrives L(x) is the rate at which sand leaves Y(x) = R(x) - L(x) is the total rate of change of sand at the beach
When do you do u substitution on integrals?
If you need the chain rule to take the derivative of a function, you need u substitution to take the integral
Do you always have to complete the square and reconfigure to take integral of something that inevitably equals an inverse trig identity?
NO strategies, 1. you can divide numbers and √x under the square to fit the format u^2 - 1 in the denominator, or anything related 2. you can take out u as x^2, so if it is x^4 under the numerator, it goes to u^2 instead, and matches format 3. you can factor into trig identities 4. you can factor out trig terms, sinx, cosx, to make a du value or trig ratio be easier to factor ** 5. you can split trig terms into 2 ratios instead of one, making it into more recognizable, manipulatable forms
IN accumulating rates, how do you use calculator/find values involving total rates?
Plug in both rates into Y1 and Y2 subtract/add y2 and y1 to craft total rate, depending on which rate causes a positive ROC to the item, and which should be subtracted bc of negative ROC can take integral of Y2-Y1 (or any other examples) AND find zeros, etc
How do you find integrals that give you inverse trig values? d/dx of arctan(x) arcsin(x) arcsec(x)
Reconfigure the integrand to fit the 3 inverse trig derivatives arctan = 1/x^2 + 1 arc sin = 1/√1 - x^2 arcsec = 1/√x^2 - 1
Why should u always check callout of question?
SOME Qs WILL ASK YOU TO JUST WRITE INTEGRAL, NO SOLVE KIDDO
How to take derivative/integral of functions with log/ln? What types of functions do u use ln differentiation?
Split up with log rules ALWAYS IF U CAN - long snotty fraction functions, but always 2 - transcendental functions - MUST split up to take ln on both sides for x^x
How to find abs max/min that accumulating rates frqs often mention? Justification?
Use Candidates Test - check endpoints of interval given - check cvs Find Cvs? set total rate equal to 0 for first derivative test and graph. Look only at CVs that indicate max/min: where first deriv changes from + to - or v.v. Plug in values into total amount function, and solve for max/min value Justification?
When do you do long division to solve integral? how?
Use long division when power in numerator of integrand is greater than or equal to power in denominator - if x^2 in denom, do polynomial long division, or do synthetic if linear divisor (stick to long division buddy) 1. do long division, split integrand into quotient + remainder/divisor 2. take 2 integrals, one of quotient value and one of split up remainder/divisor - keep splitting up second/complex part of integral into many integrals until they are easier to solve - use u substitution/any other method to simplify down complex integral 3. solve out both integrals and add - +C if indefined - number ONLY if defined
How to take a u substitution and cancel out a ratio like tanx with du? ex: ∫tan(x) ln(cosx)
Workflow: long division doesn't work, nor does completing the square, this won't format into an inverse trig derivative Take u, which derivative can change into tanx u = lncosx du = 1/cosx (sinx) = tanx - this ratio manipulation can cancel out the tanx with du
If a function is an odd function, what does this mean for it's integrals?
a ∫ f(x) dx = 0 -a
If a function is an even function, what does this mean for it's integrals?
a a ∫ f(x) dx = 2 ∫ f(x) dx -a 0
given function v(t) average v(t) on interval average a(t) - ROC of v(t)- on interval
average v(t) = 1/b-a ∫v(t) dt average a(t) = v(b) - v(a) / b - a
when there is a 1/x and lnx in the integrand, what should you choose as u? why?
choose u = lnx bc du = 1/x this will cancel out the 1/x in the integrand for the du value
A definite integral produces what type of answer? So what is the derivative of any definite integral?
definite integrals produce a constant as an answer deriv = 0 of a constant
What does differentiate mean? vs. take the differential
differentiate - take derivative differential - move dx, only ROC of dy
what are even and odd trig identities?
even: cos(x) = cos(-x) sec(x) = sec(-x) odd: sin(-x) = -sin(x) csc(-x) = -csc(x) tan(-x) = -tan(x) cot(-x) = -cot(x)
when 2 rates are present (adding/subtracting), what is the function for the total amount of ____ present at any given time? Given initial value Total rate of change
initial value + ∫A(t) - R(t) dt Rate of Change: A(t) - R(t)
∫sec u du
ln | secu + tanu | + C
∫cot u du
ln|sinu| + C
All deriv rules: d/dx of: product quotient e^u trig (sin, cos, tan, cot, sec, csc) inverse trig (arc - sin, cos, tan, cot, sec, csc) chain log a u ** ln 2^x
log a u = 1/lna (1/u) (du)
When doing u substitution of indefinete integral, what 2 things must u do?
plug in u value, to be in terms of x again add +c, the arbitrary constant is still there
If an FRQ/ question says that sand arrives at a constant rate of 8 pounds per second, what do you put in the total rate?
rate of arrival is JUST 8 8 -√t+1 - constant rate, not linearly increasing kiddo
when the argument is positive for a definite integral, with ln|x|, what can u do in answer of ln?
remove abs value signs
How do you take the integral when du is not already in the integral?
set du equal to the value inside the integral that may translate to dx. You can replace du in the integral with any constant multiple/coefficient that equates it to the dx value in the integrand. Solve, taking constant multiple out of integral OR, multiply by a constant multiple in the integral which will take out the coefficient from the du substituted value. This will equate it to du and leave with you with the same constant multiple factored out
How do trigonometric functions and their derivatives effect what values you should take as u?
set u equal to the trignometric function and simplify the integrand so that du = the derivative of the trig function, and the integrand has that derivative which can be factored out ex: ∫3^tanx/4cos^2(x) dx u = tan x du = sec^2x = 1/cos^2x , can cancel out in integrand
how do u take the integral of cot(u), and other more complicated trig ratios, without the rule?
simplify cot(u) into it's ratio, cos(u) over sin(u) - use u substitution and du to cancel values out, getting ln|sinu| + C
What does the integral of a rate give you?
the net change in what the rate is measuring
How to graph total rate equation for accumulating rates frqs? what can this help with?
turn off y1 and y2 graphing - enter on = sign plug in y3 = y2 - y1, or wtv total rate equation is graph y3 ALWAYS CHANGE INTERVAL OF X TO ONLY INTERVAL BEING OBSERVED
How do we solve complicated integrals, where there is an inside and outside function?
u substitution
How/when do you use completing the square as a method of integration analysis? How can u predict what it will simplify to?
when u substitution cannot be used to simplify, and denom is greater degree than numerator - completing the square simplifies down to an inverse trig function, depending on how the function looks Ex: 1/x^2 + 2x + 4, since the denominator has no root, and cannot be any other method, by completing the square, this simplifies to arctan, bc no root 1/√x^2 - 4, since the denominator has a root, but no value outside, and greater power than num/no u sub, must complete the square to become arcsin 1/x+2 √(x+4x+4), since the denominator has a root multiplied by a term outside, and greater denom power/no u sub, must be a completing the square to become arcsec
integration rules ∫a^u
∫a^u = a^u/lna
