Indefinite Integration Questions and Answers

Calculus

Indefinite Integration

Find the Taylor series for f x centered at the given value of a Assume that f has a power series expansion Do not show that R x 0 f x sin x a f x 8 n 0 R fr TO n X T n X Find the associated radius of convergence R

Calculus

Indefinite Integration

Use the definition of a Taylor series to find the first four nonzero terms of the series for f x centered at the given value of a Enter your answers as a comm separated list 7 7 3 f x x 2 7 1 x 7 9 27 a 2 X 2 3 2 T 7 81 x 2 3 X

Calculus

Indefinite Integration

Write the given function as a power series cos x 1 X n 1 1 n 1 1 X Evaluate the indefinite integral as an infinite series cos X 1 1 x2n 2n 2n x2n 2n 2n dx C c

Calculus

Indefinite Integration

We can now solve 1 u Ke 2 2t for u 1 Notice that since u 1 is a solution This also means that K could be 0 and so we could simply write u

Calculus

Indefinite Integration

Part 5 of 6 We now have In 1 ul 11 2 2 2t C 2 2t C This can be re written as 11 u e 2 2t C Letting K e e can be re written as K

Calculus

Indefinite Integration

Part 3 of 6 Integrating the left hand side and ignoring the constant of integration gives us remember to use In lvl where appropriate In 1 u In Ju 1 Part 4 of 6 1 u du Integrating the right hand side and ignoring the constant of integration gives us t dt Submit Skip you cannot come back

Calculus

Indefinite Integration

Seperating the expression in u from the expression in t we get 1 u 1 Part 3 of 6 1 1 u du 2 t dt Integrating the left hand side and ignoring the constant of integration gives us remember t du 1 u

Calculus

Indefinite Integration

1 of 6 dt separate du 2 2u t tu factor the right hand side to get dt du dt 2 t

Calculus

Indefinite Integration

Using the substitution u x2 88 and again ignoring the constant of integration gives us du

Calculus

Indefinite Integration

On the right hand side we have X 12 x2 88 dx This can be done by using the substitution u and du dx

Calculus

Indefinite Integration

Evaluate the integral Remember to use absolute values where appropriate Use C for the constant of integration 12 3 dt t 9

Calculus

Indefinite Integration

Write out the form of the partial fraction decomposition of the function as in this example Do not determine the numerical values of the coeffi a 6 b x2 4 A x 2 8 x x 1 x 8 B A X 12 2 B x 2 x 12 2 C x 8 7 4 006 2

Calculus

Indefinite Integration

Write the given function as a power series 2 C 1 7 Evaluate the indefinite integral as an infinite series cos x 1 dx cos x 1 1 c

Calculus

Indefinite Integration

2 Use Stoke s Theorem to evaluate SF d C F x y x 3zsin x x e y cos z S z 1x y z 0 Z

Calculus

Indefinite Integration

13 x t 3 y Answer x 3 t 3

Calculus

Indefinite Integration

stion 24 A hot air balloon is flying at a height of 152 feet and is directly above Marshall Space Flight Center in Huntsville Alabama THe pilot looks down at the airport which is known to be 5 miles from Marshall Space Flight Center What is the angle of depression in degrees for the baloon to the airport Round to the tenth

Calculus

Indefinite Integration

1 Compute the value of f x 7 3xy 5 dr if C is the curve consisting of the circle centered at the origin of radius 3 and the line y x as depicted in the picture 3

Calculus

Indefinite Integration

If sin x in Quadrant 1 find 27 28 tan 2x Please enter answer accurate to 4 decimal places

Calculus

Indefinite Integration

Directions Draw a diagram create an equation and then solve for the desired measurement Kami is standing 12 feet from a rock climbing wall When she looks up to see her friend ascend the wall the angle of elevation is 56 How high up the wall is her friend V HINT Round to the nearest thousandths if necessary

Calculus

Indefinite Integration

Given AABC below with m B 100 a 6 and c 12 find the area of the triangle Round your answer to the nearest tenth and do not include units in your answer C B 100 b 12 A

Calculus

Indefinite Integration

Find the missing side length and angles of AABC given that mZB 34 C Select the correct answer below B b 12 0 A 44 4 and C 101 6 b 11 0 A 444 and C 101 6 b 12 0 A 434 and C 102 6 Ob 110 A 43 4 and C 102 6 34 15 21 b

Calculus

Indefinite Integration

nd the most general antiderivative of the function Check your answer by differentiation Use C for the constant of the antiderivative f x x 9 2

Calculus

Indefinite Integration

Now we must find f x which is given by the general antiderivative of f x 18×2 C We note that f x is the sum of two functions h x 18x and b x C such that f x h x b x This means that f x will be given by the following where H x is the general antiderivative of h x B x is the general antiderivative of b x and D is an arbitrary constant f x H x B x D Furthermore we note that h x 18×2 is the product of a constant and a function k x x such that h x 18k x Finally we note that b x C is a constant function and can be written in the form b x Cx Applying the formulas described in this step and those used in the previous step we can now state the following letting D represent the arbitrary constant Do not add any additional constants f x 18x C 18 f x 18 Cx D To conclude give an equation for f Use C for the constant of the first antiderivative and D for the constant of the second antiderivative f x

Calculus

Indefinite Integration

The values of y yo and t are given For an equation of the form y yoekt give the exact value of k in terms of natural logarithms Yo 120 t 3 y 40 k 0 se integers or fractions for any numbers in the expression www

Calculus

Indefinite Integration

Problem 4 20pts Use the Divergence Theorem to evaluate the following Surface Integral fr where the vector field F is given by F dS F 2x 2 i x y j xz k and the closed surface S is given by the unit cube where the coordinates x y and z take values from 1to1 raspectively

Calculus

Indefinite Integration

Consider the following function f x 36x Find f x Use C for the constant of the first antiderivative f x Find f Use D for the constant of the second antiderivative f x

Calculus

Indefinite Integration

Find an antiderivative of the function Remember to use absolute values where appropriate 5 t a g t G t b r 0 sec 0 R 8

Calculus

Indefinite Integration

We have determined that w 22 Recalling from a previous step that 44 w allows us to determine the other dimension 1 44 w To conclude we state the dimensions of the rectangle in m with perimeter 88 m whose area is as large as possible Enter the dimensions as a comma separated list m

Calculus

Indefinite Integration

IF bao lun 848 and Couluate 1 b lon 31 A The series will absolutly converge B Notenough informaler The series will Conditionally convere 5 The serves will

Calculus

Indefinite Integration

Use a power reducing formula to simplify 32sinacos a Write your answer as an equivalent expression that does not contain any powers of sine or cosine greater than 1

Calculus

Indefinite Integration

Find the solution of the differential equation that satisfies the given initial condition dP dt Pt P 1 2

Calculus

Indefinite Integration

You are given that tan A 4 and tan B 7 Find tan A B Give your answer as a fraction

Calculus

Indefinite Integration

2 Evaluate xsin 8x dx

Calculus

Indefinite Integration

Evaluate cos 4x sin 4x dx

Calculus

Indefinite Integration

1 Evaluate x In x dx

Calculus

Indefinite Integration

S 1 sin x x1 2 dx

Calculus

Indefinite Integration

Along the curve r t t i j tk 0 st 1 evaluate each of the following integrals a x y z dx b x y z dy x y z dy c f x y z a b C C C K y z dx 13 6 CA C f x y z dy z dy Type an integer or a simplified fraction C Type an integer or a simplified fraction dm X

Calculus

Indefinite Integration

Find the line integrals of F 4yi 2xj zk from 0 0 0 to 1 1 1 over each of the following paths a The straight line path C r t ti tj tk 0 st 1 b The curved path C r t ti t j t k 0st 1 c The path C3UC4 consisting of the line segment from 0 0 0 to 1 1 0 followed by the segment from 1 1 0 to 1 1 1 0 0 0 C 1 1 CS CA 1 1 0 a The line integral of F over the straight line pa C is Type an integer or a simplified fraction

Calculus

Indefinite Integration

Given the side length b 15 and the angle B 26 on the triangle below find the lengths of a and c and the measure o angle A Do not round during your calculations but round your final answers to one decimal place 15 C Provide your answer below A 26 a B

Calculus

Indefinite Integration

Find the indefinite integral Check by differentiation 12e 10 dz

Calculus

Indefinite Integration

6 Prove the following 1 cosx sinx cosx 1 a sinx cosx sinx cos x 1 cos x b sin x 2csc x 1 c cos x cotx cot x cos x 1 sinx secx tanx d 1 sinx e sec x tan x tan x sec x

Calculus

Indefinite Integration

Determine whether the improper integral converges or diverges If it converges find its value da You must show your work to receive full credit Edit Format Table

Calculus

Indefinite Integration

Use Half Angle or Double Angle Formula to find f sin 5xdx You must show your work to receive full credit

Calculus

Indefinite Integration

Use partial fractions to evaluate d dx O 5 ln 2 1 6 C O In x 2 x 3 C O In C O In C 5 x x 6

Calculus

Indefinite Integration

Use Integration by Parts Formula to find e dx Or e 2re e C O e 2re e C e 2xe 2e C

Calculus

Indefinite Integration

Use a substitution to find f sec 0 tan de sec 0 Os C O4 sec 0 tan0 C

Calculus

Indefinite Integration

Use a substitution to find fre 1 da O e 1 2x e C 0 0 O e 41 C

Calculus

Indefinite Integration

Use a substitution to evaluate the definite integral cos x da Osin 1 Ocos 1 sin 1

Calculus

Indefinite Integration

Given S is a surface for a part of the sphere x y z 4 that lies above the cone z x y Find a parametric representation for the surface S in term of o and 8 Then determine the domain of and 0 Based on the expression written previously show that rx rell 4 sin o Hence compute the surface integral of ffs ds where the surface S is given parametrically by r 0 0 Let C is the boundary curve along the bottom of the upper half of the sphere thus use Stokes theorem to compute the line integral of fo F dr where F x y z xzi yzj xyk and upwards orientation

Calculus

Indefinite Integration

Differentiate f x sin x cot x 4 F x cos x csc 2 x