Indefinite Integration Questions and Answers

sin 9x dx x cos 9x 9 sin 9x 81 C

f x x 2 e 9 dx 9x 19 9x 81 e C

This problem asks you to evaluate using integration by parts Fill in the table below with the appropriate terms for setting up the integration by parts u Complete the last integration to get 12 du x ln x dx dx V dv Now substitute all of this into the Integration by Parts formula giving x 1 x dx In o do dx 0 10 C da

Consider the integral methods 4x 4x4 5 1 dx In the following we will evaluate the integral using two A First rewrite the integral by multiplying out the integrand 9 4 4x4 x 1 dx 4x 4x da Then evaluate the resulting integral term by term 4x4 4x4 xr5 1 dr 10 4x4 x5 1 dx 2x 5 4x4 x5 1 dx x 10dw 5 4x 5 B Next rewrite the integral using the substitution w x5 1 424 25 c Evaluate this integral and back substitute for w to find the value of the original integral 43 C How are your expressions from parts A and B different What is the difference between the two Ignore the constant of integration answer from B answer from A Are both of the answers correct Be sure you can explain why they are

Evaluate the integral using an appropriate substitution x x8 5 dx C

19 a Find and interpret C 15 20n if 1 n 30 if 30 n 45 C n 15 50n 10n if n 45 21 b How much does it cost to order 45 copies of the book What about 46 copies 22 Your answer to 19b should get you thinking Suppose a bookstore estimates it will sell 45 copies of the book How many books can in fact be ordered for the same price as those 45 copies Round your answer to a whole number of books 0 An on line clothing retailer charges shipping costs according to the following formula 55 SC

Question 5 Consider the function y f x graphed below Give the a a intervals on which the derivative of the function is negative x b intervals on which the value of the function is negative x c one x value where the derivative of the function is the least meaning most negative x d one a value where the derivative of the function is zero x

Evaluate the integral sint dt 5 3 cost OA 1 4 3 cost 4 OB 4 3 cost OC C 1 3 cost 4 OD C C 1 6 3 cost 6 C

Evaluate the integral 3x 1 10 2x dx OA 12 10 2x3 5 4 C OB 2 10 2x 3 4 C OC 3 10 2x3 5 4 C 2 OD 10 2x 5 4 C C

3 64a 6 39 bc 6 64a 39 bc Simplify your answer Type exponential notation with positive exponents

a Find the critical point of f x y 2x y 2 4y x y b Find the discriminant D x y for the function D x y what is the value of the discriminant at the critical point D What does this tell us about the critical point The discriminant doesn t tell us anything c Sketch contours near the critical point to determine or confirm whether it is a local maximum a local minimum a saddle point or none The critical point is a saddle point

13 Using the washer method determine the volume of the solid obtained by rotating the region bounded by y 2 x 1 and y x 1 about the line x 1 1 0 1 2 3 7 6 5 4 3 1 2 3 4 5 X

Is point B a local maximum local minimum or neither A local minimum B local maximum C neither A B

The voltage across a 4 5 F capacitor in a copying machine is zero What is the voltage after 19 ms if a current of 45 mA charges the capacit The voltage at t 19 ms is V Type an integer or a decimal

Evaluate the following indefinite integral 30x dx 2 3 2 30x dx

Evaluate the following indefinite integral 120x dx 2 0 dx X

Determine the following 12x dx 12x dx

Find an antiderivative of the given function f t 6t 12 What is a possible antiderivative of the given function Select the correct choice belo 3 4 t 12t O A F t 2 OB F t 6t4 12t O C F t 6t4 12 OD F t t4 6t

b Given two polar curves 3 cos and 2 3 sin such that 0 0 2r i Find the intersection point s ii Sketch and on the same diagram iii Find the arc length of r that lies inside 2

If f x sin there exists a number c in the interval x 3 that satisfies the conclusion of the Mean Value Theorem Which of the following could be c O A 2T B 5T C T D 3 4 3 2

Which of the following graphs is consistent with the differential equation dy dz 2 O A B C D E Vi

X 0 9 1 0 1 1 Given h x 5 3 5 0 4 7 evaluated at x 5 O A 3 OB 3 O C 1 O D OE 3 what is a reasonable value forh x

X O A B M dx 23 2 C 2 23 2 C C 223 23 2 C O D In 23 23 2 C O E x In 23 23 2 C

Hint try the substitution u x O A arcsin x C OB B arcsin u C c arcsin x C O C D x arcsin x C O E 2 arcsin z C

3 The base of a solid is a right triangle whose base side has length a and whose perpendicular side has length a Find the volume of the solid if cross sections perpendicular to the base of the triangle are semicircles

2 Two separate rabbit populations are observed for 80 weeks starting at the same time and with the same initial populations The growth rates of two rabbit populations are modeled as follows where t 0 corresponds to the beginning of the observation period ri t 4 sin t 0 11 1 where r is rabbits per week and is time in weeks r2 t t 2 where r is rabbits per week and is time in weeks Below is a graph of the curves representing the rates of growth of the two populations over the observation period 10 20 30 50 60 70 80 A Using your calculator find approximately the first positive time for which the rates of growth for the two populations are the same B What s the physical significance of the area between the two curves from time t 0 until the first time where the two rates are the same What does the area represent I C Suppose you want to find the first time call it 7 after the beginning of the observation period at which the two rabbit populations have identical populations Write an equation to solve for the unknown variable T

lowing is not true OA Figuring the volume of a solid can involve slicing it into cross sections B In a solid of revolution the cross sections are circles or washers C A solid of revolution is obtained by revolving a region about a line D You can figure the volume of a solid by slicing it into cross sections figuring the individual volumes and adding them up E You can figure the exact volume of a solid by slicing it into cross sections figuring the individual volumes by multiplying cross sectional area times thickness and adding them up

A Find the exact area of the region in the first quadrant bounded by the curves y f x 2x 1 and y x Exact area means no calculator numbers B Find the inverse function y f x C Using part A and the notion of symmetry between a function and its inverse find the exact area of the region in the first quadrant bounded by the curves y f x and y x Explain your reasoning Hint Think graphically and little or no math will need to be done D Find a value for a such that the average value of the function f x on the interval 0 a is equal

Powen to the following differential equation you do not senies to find the powen senies solution need to find the nadius do not need to find the familiare function You of convengence You Use Powen senies to solve and express final solution as a power senies won k must Your all

Estimate the price of one barrel of crude oil on January 25 2016 and the rate at which the price was rising on that day Your answer for the rate should be in dollars per day The price of one barrel of crude oil on January 25 2016 was about Type an integer or a decimal rounded to two decimal places as needed Barrel Price of Crude Oil 30 29 28 27 26 25 24 23 0 5 10 15 20 25 30 January 2016 o o

The definite integral of f x from a to b is defined to be O A f f x dx limn n f wh k O B f f x dx limn k 1 wk A k a 1 c f f x dx F a F b b D f f x dx limn 1 Ax wk k a k n ab O E f f x dx F b F a a

Factor the polynomial 2 8x 16 8x 16

Careful here O A 1 OB 1 OC 0 OD 3 O E None of the above

The point P xo yo zo is in a plane for which n a b c is a normal vector Which of the following equations must every other point Q x y z in the plane satisfy Select all that apply n PQ 0 1x PQ 0 nx ax by cz d where d axo byo czo a b c x xo y yo 2 zo 0 a x xo b y yo c zzo 0

sin x dx O A 4 7 5 tan x C O B 7x 2 7 5 tan x C O c 4 7 5 cotx C O D 4 7 5 cotx C OE 7x 2 7 cotx C

6 Below is the graph of y f x on the interval 0 8 A Based on the graph put the following in order from least to greatest xxx xxx xxx xx and f B Let A represent the average value of the function f x on the interval 061 Is there a val

Find any local maximum or minimum points of the given function y x 4x 3x 1 Select the correct choice below and if necessary fill in the answer box to complete your choice OA The local maximum point s is are Simplify your answer Type an ordered pair using integers or fractions Use a comma to separate answers as r

Find da 9 x for g x sin x 5x OA sin x 20x OB sin x OC cos x OD sin x 120 OE cos x 120 x

approximately what values of x between 5 and 5 does the original function y have inflection points W M 3 OA x 2 03 x 0 x 2 03 OB x 2 03 x 2 03 OC x 3 83 x 3 83 OD x 4 91 x 2 03 x 0 x 2 03 x 4 91 O E There are no points of inflection

Evaluate the following definite integral to three decimal places by 2 interpreting it in terms of areas 9 x 1 2 dx 1 OA 7 0686 OB 34 274 O C 13 069 OD 20 137 OE 8 3562

181 consists of two straight lines and a semicircle y f x 2 Use this graph to evaluate to three decimal places f f x dx OA 14 000 OB 12 000 C 34 566 D 28 283 OE 20 283

2 For each of the functions below use your graphing calculator to draw a graph of the function and then estimate the x coordinates of its inflection points List all estimated points of inflection all intervals where the function is concave up and all the intervals where the function is concave down You don t need to show your graphs and of course your estimates don t have to be perfect A f x xe B f x tanx on the domain x C f x 1

b Let f R R be a function and suppose ff t dt a Show that x y z b y 2 where D is the unit ball x y z x y z 1 dv 4

Find the indefinite integral f sin x cos adx O cos zsina C cos a sin a C O 0 240 O C

Find the indefinite integral f x da Please show your work to receive full credit

Find the Fourier series for the periodic function x 0 2 x if 1 x 0 if 0 x 1

2 Find the antiderivative 0 4 0 17 S 2 2 3 dx

2 In 2 1 dz 0 1 0 0 5 0 C 2 1 32 422 02

4z 32 1 Find S 0 0 24 6 32 1 dz C C C 9 32 11 C

gchool Find the general indennite intergal Land Sin 20tdt Sin 10 t A shot C 5 6 cas 10t C Sin lot C