Indefinite Integration Questions and Answers

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

Evaluate the following indefinite integral 1 88 1 In 65 ds H help formulas

2 1 4 Pts Given the velocity history given by t 6e t cos t 4t e and the initial condition 0 2 1 3 find an expression for r t JS CO you integrate t to be come F t 3 7 4 S 6e cos 4 et at 3 t 32e 4 sin t t et 6 10

2 Determine whether the infinite geometric series converges If it does find its sum 1 1 1 64 32 16 a 8 c 3 1 4 i 4 4 b 4 1 3 9 27

Find the volume of f x y x 3 3y 2 above the 2D region bounded by y x 2 y x 2 and the y axis Give both orders of integration

For the following exercises use shells to find the volumes of the given solids Note that the rotated regions lie between the curve and the x axis and are rotated around the y axis 120 y 1x x 0 and x 1 121 y 5x x 0 and x 1 122 y 1 x 1 and x 100 123 y 1 x x 0 and x 1 1 124 y 12 x 0 and x 3 1 x2 125 y sinx x 0 and x T 126 y 0 and x 1 2 2 127 y x x 0 and x 1

The amount of electricity consumed in kWh on a randomly chosen day in a warehouse can be modelled by the random variable with probability density function given by for 30 x 150 f x otherwise Find the probability correct to 2 decimals that on a randomly chosen day more than 115 kWh of electricity will be consumed Probability rounded to two decimals

Because of the relatively high interest rates most consumers attempt to pay off their credit card bills promptly However this is not always possible An analysis of the amount of interest paid monthly by a bank s Visa cardholders reveals that the amount is normally distributed with a mean of 29 dollars and a standard deviation of 9 dollars A What proportion of the bank s Visa cardholders pay more than 31 dollars in interest Proportion B What proportion of the bank s Visa cardholders pay more than 38 dollars in interest Proportion C What proportion of the bank s Visa cardholders pay less than 17 dollars in interest Proportion D What interest payment is exceeded by only 22 of the bank s Visa cardholders Interest Payment

11 5 Use the information given to find the exact value of the angle A Leave you answer in the exact form A a 2 b b 4 C a B C 5

x a If F x 2 1 dt then F x F 12 b If F x c If F x d If F x 4 17 t 1 t x 1 dt then F x 2 cos x dt then F x 1 F t F dt then F x help formulas

e5x 7 dx

The cost to attend a public university in a recent year is 14 799 The circle graph to the right shows the percentage of that cost for tuition fees room board and computer costs Determine the cost in dollars for each category The cost of tuition and fees is Round to the nearest cent as needed Cost to Attend a Public University Tuition Fees 33 3 Room 36 3 Board 23 6 Computer costs 6 8

4 Evaluate y x x y where E is the region outside the region enclosed by the ellipse Se E x y 16 9 dxdy 1

ind the Laplace transform of the following function f t f cos tr e sin 7 dr

4 x n 1 Find the sum of the series for those values of x Need Help Read It 2 2 Points 00 DETAILS n 1 Find the values of x for which the series converges Enter your answer using interval notation Watch It x 7 SCALCET9 11 2 060

Evaluate y dr both directly and using Green s theorem where y is the semicircle in the upper half plane from R to R Solution Evaluate y de directly set y 1 7 2 S y dx Lydx ya y 0 x R cost y R sint dx dt dy 0 dt y dx SS dr dx Rsint dt dy R cost dt TR dy 2 dy 2 dy 7 2 ff dx dy dx dy R 0 dt 0 R 0 t n R t R Now we evaluate fyda this time using Green s theorem In this case P x y y and Q x y 0 y r sin 0 x r cose dx dy rdr d0 0 r R R TR 0 0 T Rsint Rsint dt R th r dr de r dr d0

A 200 gallon tank initially contains 100 gallons of water with 20 pounds of salt A salt solution with 7 10 pou of salt per gallon is added to the tank at 10 gal min and the resulting mixture is drained out at 5 gal min E Q t denote the quantity lbs of salt at time t min a Write a differential equation for Q t which is valid up until the point at which the tank overflows Q t b Find the quantity of salt in the tank as it s about to overflow

aluate the double integral foudyda Show all steps Do not reverse the order of integration

Il in the blank with an the corresponding sides of two similar figures are the same length the figures are called If the corresponding sides of two similar figures are the same length the figures are called figures figures congruent regular proportional

For which values of aeR does the following integral converge sin x xa dx

Find the indefinite integral 12x cos x 27 dx O3sin x 27 c O 3x sin x 27x c 3x sin x 27x c O 3sin x 27 c

Solve y 4 x 24e 6x 9sinx given that the solution passes through the point 0 8 8 O y X 4e 6x 9cosx 21 N W Oy 2 2 Oy 6x 4e 6x 9cosx 21 8 2 x 4e 6x 9cosx 3 3 2 O y 6x 4e 6x 9cosx 3

Find the indefinite integral eix 7x 36 dx 7x On e x 36 c ex e ex 36 c O 7eXIn ex 36 C 7 n e7x 36 c

Find the flux of the vector field F 0 0 3 across the slanted face of the tetrahedron z 4 x y in the first octant Normal vectors point upward Set up the integral that gives the flux as a double integral over a region R in the xy plane SSF n 15 SSC S R Type an exact answer Fonds dA

Solve y 18x 12e the point 1 3 Oy 6x 2e 6x 6 Oy 6x 2e 6x 6 6 9 given that the solution passes through 6x 6 11 3 Oy 6x 2e 6x 6 gx 2 Oy 6x 2e 6x 6 7 6x 6 9x 2

Evaluate the following definite integral 6 S 18x x 9 dx O 0 O S18x x 9 dx 13392 6 3 18x 18x x 9 dx 13500 6 0 18x X 9 O S 10 18x x 9 dx 13500 o s J 18x 0 2 3 18x x 9 dx 9 dx 13392

Find the following indefinite integral 15secxtenx ax O 15tanx 24x c 1 1 0 O15secx 24x c 4 Stam 1 O 15secx 6x 0 1 dx

Find the following indefinite integral 4 156 O 15e 4 In x 3cosx c O 15e O 15e 3sinx dx 2 X 2 X 3cosx c 3cosx c 15e 4 In x 3cosx c

Shown below is the graph of the flow rate f of a pipe In gallons per minute Here the number of minutes Using this information find Velocity Gallons Minute gallons of water that howed through this pipe in 7 minutes E 2014 17 gations S Luna 1 grana at which gives the total number of Time in Minutes

Find the following indefinite integral 5x 5x sin x4 12 dx Ox cos x512x c X COS 5 4 cos x 12 c 5 O cos x 12 4 0 2x cos x512x c

Find the values of sint cost tan t csct sect and cott if P 5 2 3 is the point on the unit circle that corresponds to the real number t GEES sint Simplify your answer including any radicals Use integers or fractions for any numbers in the expression

Find the following indefinite integral S 12x 8 15x 27x4 3 5 O 4 5 5 x 5x 9 In x c 6 10 0 5 O 4 5 6 10 5 9 In x c 5 6 27 5 5 27 LOX 5 3 4x 4 1 C

Find the indefinite integral f 2x 9 x 9x 12 5 dx 02 x 9x 12 5 C 26 O O 25 x 9x x 9x 12 6 c 21 x 9x 12 26 0

Simplify the following expression x dx d dx dt dt 1 2 1 1

Find the area of the region R bounded by the graph of f and the x axis on the given interval Graph f and show the region R f x x x 10 1 11 The area is Choose the correct graph below OA Round to the nearest hundredth as needed 200 Q E All graphs shown in a 2 12 1 by 200 200 20 viewing window B GOCLE Q O C Q

and simplify the derivative of the following function f x 3x5 2x 1 Which of the following shows how to find the derivative of f x d O A f x x 3x 2x 1 3x 2x 1 dx dx B f x OC f x 2x 1 3x 3x 2x 1 dx dx 2x 1 3x 2x 1 2x 1 x dx dx 3x5 D f x x 3x 3x 2x 1 2x 1 dx The derivative of f x 3x5 2x 1 is

Find the indefinite integral 6 sec x tan x 7 sec x dx S 6 sec x tan x 7 sec x dx

Determine the following indefinite integral 20t 8 t 201 8 t dt dt

the area of the region bounded by the graph of f x x sin x and the x axis between x 0 and x Determine a change of variables from x to u Choose the correct answer below OA u xsin x B u sinx OC U X OD u x Write the integral in terms of u xsin x dx 0 Jo du The area of the given region is Simplify your answer

Evaluate the following indefinite integral S 10 10 x dx 10 10 x dx 0

Evaluate each of the following integrals 3x a S x dx 2 1 9x

Use the substitution u 2x 3x to evaluate the indefinite integral below 4x 3 2x 3x dx Write the integrand in terms of u JO 4x 3 2x 3 2x 3x dx Evaluate the integral S 4x 3 2x 3x dx du

for each of the following iterated integrals draw the region of integration and find the value of th ntegral 1 y SS xz dydxdz

5 Solve on a calculator but show work cot 5

Let F x y and C be the circle of radius 6 centered at the origin oriented counterclockwise Evaluate parameterizing C Use a parametric description of C and set up the integral F 2x dr 0 dt S F dr F

The velocity function is v t t 6t 8 for a particle moving along a line Find the displacement and the distance traveled by the particle during the time interval 1 5 displacement distance traveled 18 66 If needed see page 287 of the Stewart Essential Calculus textbook for the definitions of these terms

Find the exact value of the integral using formulas from geometry 5 5 2 x dx 3 2 x dx 3 Simplify your answer

Use substitution to find the indefinite integral 2 42 5dz 2 42 5dz

Use substitution to find the indefinite integral 3x 0x dx 3x 6x dx

Use substitution to find the indefinite integral 56 118 dx 0 8x 1 8 8x 1 8 dx