Indefinite Integration Questions and Answers

Find the most general antiderivative of the function for 18x 4x 9 F x 6x 14x C BI x 3bx 14 C F x 6x 7 9x c F x 18x 14x 9X C

Evaluate the integral by making an u Substitution x 2 dxy 312 13 x 2 C B Ya x 2 2 C xXx 7 2 1 2 C 11 11 1 C 312 3 X

Eval Ssin x cos cos x dx Bin casx 2 C 3 2 cos sinx C 3 2 sin los x C wants

Evaluate the indefinite intera S 4 10X 7 4x 5x 2 2 A 2 7 4x 5x C B 7 4x 5x C C 2 7 4X 5x C

Now if u x 30 then 3 x x 30 dx u 2 1 1 du u 2 du x x 3 3 This evaluates as follows Enter your answer in terms of u 13 01 20 du C

Use substitution to find the indefinite integral 6 fox 5x x ax e dx Sox 6x59x6 dx

Evaluate the cylindrical coordinate integral The value is 0 0 r r dz dr d

Make a substitution to express the integrand as a rational function and then evaluate the integral Use C for the constant of integration 7e2x 11e 18 e2x dx 9 ln e 9 2 ln e 2 C X

Evaluate the integral 122 2 1 2x 3x 1 0 2 ln 3 In 2 dx

Evaluate the integral Remember to use absolute values where appropriate Use C for the constant of integration y 2 5y 1 dy In y 2 In 5y 1 C 11 X

Write out the form of the partial fraction decomposition of the function as in this example Do not determine the numerical values of the coefficients a x5 4 x x x 4x 4 b A B r 1 x x x 6 B A x 3 x 2 Cx E x 2 X Dx F x 2 X

olution of the differential equation that satisfies the given initial condition dA Ab cos br A 0 b dr b sin br In 6 e No REVIOUS ANSWERS SCALCET9 9 3 015

10 1 2 Points DETAILS Determine whether the geometric series is convergent or divergent n 1 convergent divergent 16 0 64 1 PREVIOUS ANSWERS SCALCET9 11 2 027 x If it is convergent find its sum If the quantity diverges enter DIVERGES 44 4444

Consider the following series 1 2 4 4 n 1 Does the function f x O Yes O No 1 x 4 satisfy the conditions of the Integral Test Evaluate the following integral If it is convergent evaluate it If the quantity diverges enter DIVERGES 1 x 2 40x dx 4 Orient finite the series is Select

Sketch the curve 2 r 2 2 sin 8 1 y 1 2 3 1 2 y X y 1 1 y 3 2 1 1 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Now we must find f x which is given by the general antiderivative of f x 18x2 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 18x2 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

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

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

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

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

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

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

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

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

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

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

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

Find the indefinite integral Check by differentiation 12e 10 dz

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

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

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

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

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

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