Indefinite Integration Questions and Answers

2 Problems 12 pts each Show all work for credit 1 Let f x 3x Use the definition of the derivative to prove that f x 6x No credit will be given for using the short cut rule

First draw a sketch of the 2D region and the kth strip Then write the Riemann Sum that will approximate the volume of revolution of the surface generated by rotating the region bounded by y 2x x 2 and the first quadrant around the x axis using the shell method

State what the first and easiest step procedure should be in finding the integral farcsin x da O trigonometric integral strategy O trigonometric substitution O Integral cannot be determined algebraically Ou substitution 4

Tutorial Exercise Use the power series 1 1 x 1 n 0 1 x Xx 1 to find a power series for the function centered at 0 Determine the interval of convergence f x arctan 2x Step 1 Use the following integration fact to rewrite the function Tax 4x 1 dx arctan 2x f x arctan 2x 3 dx

Evaluate the indefinite integral 12 9z 4 92 92 1 dz

Find the area of the region enclosed by the curves y x and 2x y 24

x and answer the questions below in seconds after the ball is hit Construct a table of values to 3 2 1 2 2 Y 32 air 32 20 36 From the table and or graph 476 425x 16x2 a Find the maximum height of the golf ball y 16 x 3 Gos 1 1 5 2 2 5 b Find the time it takes the golf ball to reach its maximum height the ground x 1 5

the function f how would you remove the discontinuity In other words how would you define f 5 in order for f to be continuous at x 4x 5 X 5 f x R

Evaluate the integral ff y z dS where S is the truncated cone z x y 1 z

Using the Intermediate Value Theorem find a closed interval a b in which the equation x In x 10 x 0 has a solution Explain your solution in English and show what you did Type your answer in the space provided

Evaluate tan 2x sec 2x dx Ita tan 2x sec 2x dx Use a capital C for any needed constant in your answer help formulas

Evaluate the following integral 1 sin 9x cos 9x 1 Use a capital C for any needed constant in your answer da help formulas

Evaluate the integral 2 1 w 2 Use a trig substitution to eliminate the root dw

Find the value of ap Answer format Enter the value Congratulations your answer is correct and saved Check Answer Save Step By Step Example Live Help

ine integral xy ds where C is the first quadrant are of the unit circle x y 1

10 Solve the differential equation y xey

In 6 Evaluate x dx 7 Evaluate x In x dx

Compute the volume of the solid obtained by rotating the region bounded by about the line x 2 V 161 6 y 3x 4x y 7 and the y axis

1 10 points Solve the initial value problem f x x 3e 4cos x f 0 2023

4 10 points Evalute the integral 1 3rdx ex 1 3ex

Use the properties of logarithms to condense the following expression as much as possible writing the answer as a single term with a coefficient of 1 All exponent should be positive 4 In In xy Answer If you wish to enter log or In you must use the keypad K Keyboard S

2 n 1 7 Find the Maclaurin series for cos x x sin x What is the radius of convergence 8 Use Maclaurin series to find

8 Use Maclaurin series to find What is the radius of convergence sin x convergence

Find the following integrals a E 2 r sin 4x dr

Given the function z cos find the domain and the equations for two level curves Show all steps

7 Sketch the curve on polar graph paper a r 0 b sin 20 c 1 coso d r 1 cos 20

gruphing calculator is required for this problem Let R and S be the regions in the first quadrant shown in the figure The region R is bounded by the x axis and the graph of y 3sinx The region 5 is bounded by the y axis and the graphs of y 3 2x and y 3sinx 5 4 3 2 1 04 1 K R 22 1 0 1 2 3 4 LO 5 X 6 Part A Find the area of S 15 points Part B Find the volume of the solid generated when R is revolved about the x axis 15 points Part C Write but do not evaluate an integral expression that can be used to find the volume of the solid when S is revolved about the x axis 10 points

Suppose g x is a function with g 1 1 and ling a 2 and f x is a continuous function Determine the value of f 1 if 4 lim x 1 In x g x x f x 8 x 2

3 Graph of a function f is shown Find all points of discontinuity of f Determine the type of each point of discontinuity of f Also on which intervals is f continuous 3 f x

f we know that f x dx In x X step you took to solve this problem x C evaluate f 1 3x f x x dr Type your soluton in the space provided and show each

Openstax Pure mathematics is in its way the poetry of logical ideas Albert Einstein SHOW WORK ON EACH PROBLEM FOR ANY PARTIAL CREDIT 10 pts each 3 Determine the definite integral dx

3 x 3x 9x 6 x 3x 2 48 6 4 2 4 y 8 2 2 4 6 x

Find the average distance from the origin to a point in region described below y y 1 x 3

Use the table of integrals to find the indefinite integral x In x dx Sxo 6 Click the icon to view a brief table of integrals

Find the flux of the vector field F across the surface S in the indicated direction 4 2 F k S is the upper hemisphere of x y z 4 direction is outward 16 OA 3 OB 413 OC 3 D

Find the indefinite integral S co cos x sin x dx Of V y sin v dv y

A heavy industrial plant located in the center of a small town emits particulate matter into the atmosphere Suppose that the concentration of particulate matter in parts per million at a point d miles from the plant is given by C 220 15d2 If the boundaries of the town form a rectangle 6 miles long and 10 miles wide what is the average concentration of particulate matter throughout the city Express C as a function of x and y set up a double integral and evaluate it The average concentration of particulate matter is parts per million

Find the least squares line and use it to estimate y for the indicated value of x X y 0 5 3 5 5 8 10 5 15 5 18 20 5 23 25 5 22 20 20 20 20 12 8 6 3 2 Find the least squares line www y Use integers or decimals for any numbers in the expression Round to the nearest hundredth as needed

Problem 1 2 points Find the solution to satisfying y dy dt 6y y 4 2

Problem 2 2 points Find the particular solution of the differential equation satisfying the initial condition y 0 6 Answer y Your answer should be a function of x Problem 3 1 point 4 dy dx y cos x 4 cos x

4 3 2 where c 21 a 18 15 12 9 6 3 The net signed area in the above graph can be written as d 4 and b d x ax b dx

Find f x 9 1 x 7 7 15 points Consider the following function f x x 3x 7x 2 a Find the first derivative b Find the second derivative c Find any values of e such that f c 0 d Find the interval s on which fis concave up e Find the interval s on which fis concave down f Find the inflection point of f

First find the interval of 0 on which the given polar curve traces the loop in the 2nd quadrant the loop that encloses the shaded leaf Then find the area of this leaf the shaded part by performing a definite integral r cos 38 2 0 5 05

Find the indefinite integral Check by differentiation 7 12 dz 1

Find the antiderivative of the following functions In x 1 f x 2 g x 3 r x X eln x X U du 10X dx dx 25 16x Sudx Suz U 1nx x U Av Se Ax Sem dx e x c X 2x 2 US 2 C

DETAILS 0 3 Sub Use a left sum and a right sum with rectangles of equal width for the given value of n to approximate the integral Round the answers to two 2 5x 1 dx n 5 left sum right sum

Evaluate the indefinite integral 7x 11 4 dx 45 V 7X 11 de dx du 1 f 7a 11 4 da 2 18 U 1 X x 1 x dx U sub with solve for x av idx Sxcus

indefinite integral this tequire separation into two integrals one need inverse trig x 5 4x 10 S dx

Find the area shown below Hint The indefinite integral of y 3x is y x C 20 10 y 3x x 2

Find the most general antiderivative of the function f x x 7 1 x where 1 x 1 Use an upper case C for any arbitrary constants F x x 7 1 x 2 C