Indefinite Integration Questions and Answers

Evaluate the follow wing indefinite integral Sa sse st dt

Graph the logarithmic function below by moving the key point y log x You can only graph integer points for example you cannot plot the point Provide your answer below 10 0 5 1 0 1 1 t 1 1

Find In 3 rounded to the nearest tenth

60 S du Jo g use the Substitution to evaluate the intergal

find the integral S 4 dx A in 1x 21 c in 1x 2 c B In G In x 2 c D5 in x 2 c

integration by par 2X ex ax 2 2 2x 2X At x x xe 3 x 1 e c 2xx the Bxext x x x Le x c 1 x e D1x x1x x c txe 2 2x Sliad 2x c XX 214 t

6 find the integral you x Ste 7 d 2 ophin A de 7 B 76 c 2 C to 7 c D C 76 c lye

1 point f x is given in the following table X 1 14 9 16 Complete the table X 1 2 3 4 no your answer 5 B f x 1 2 3 4 A BUD C type your answer C type your answer D type

Tutorial Exercise Find the indefinite integral by the method shown in Example 4 x 9 Step 1 x 9 7 7 x dx The given integral is easier to integrate Hence define U fax x 9 7 x dx Define a function in terms of u such that the term under the square root is X Differentiate the equation with respect to x du dx Step 3 Simplify the right side dx Step 2 Substitute the terms containing x in the integrand with terms containing u and rewrite the integral in terms of u x 9 7 x dx S 7 u x 9 7 x dx 1 7 u 9 u du Co Do 7 u 9 u du

Tutorial Exercise Evaluate the integral using the following values Step 1 Now use the following to evaluate the integral 1 x 8x 1 dx By basic integration rules we have and x dx 1484 x dx 28 dx 4 J5 kRx 1 x 8x 1 dx 2 Step 2 Step 3 kf x dx f x g x h x dx r x dx g x dx Therefore Then Now substitute k L 1 x 8x 1 dx x dx x3 in the equation f x dx X 3 dx x dx h x X 8 x dx and dx 1 Bx 1 dx x dx 8 x dx 1 ox 8x x3 h x dx dx

Tutorial Exercise Find the indefinite integral csc dx Step 1 Observe that 1 9 Multiply and divide the given integral csc dx by Therefore the integral becomes Step 2 1 Now g x g x 9 1 9 is the derivative of x Let g x x Step 3 csc 1 d x CSC dx Define f g x csc 1 such that f x CSC x Rewrite the given integral in terms of g x where F g x is the antiderivative of f g x csc dx 9 csc2 2 dx 9 1 The antiderivative of f x is 9 x 9 x dx 9 F g x C c x 65c x dx

Step 1 foto f x dx 3 Prox Note that f x dx f f x dx

Find an equation for the function f that has the given derivative and whose graph passes through the given point Point f x Derivative f x 2x 8 x 2 5

Complete the table by identifying u and du for the integral g x g x dx S tan x 2 sec x dx U u g x du du g x dx dx

value of the integral Express your answers correct to four decimal places 1 Trapezoidal Rule Simpson s Rule exact value dx n 4

Evaluate Assume x 0 Check by differentiating S5x In S5 5x In x dx 5x In x dx

Find the indefinite integral Stan u 13 du

1 2 Jsx V1 x 2 0 dx OA 2 25 4 1 OOB 4 2 1 O C 4 25 4 1 0 5 OD 25 4 1

Evaluate using integration by parts x 4 e 4x dx OA x 4 4x e B x 4 e 4x 1 16 1 16 e 4x C e 4x C OC x 4 e 4x e4x C OD 4 x 4 e 4x 16 e4x C

2 Evaluate Assume u 0 when in u appears X 5x 3 dx O A 25 In 5x5 3 C 1 OB In 5x5 3 C O c 25 In 5x5 3 C 1 25 In 15x 3 C O D

Evaluate Assume x 0 Check by differentiating 22x In 2x In x dx 22x In x dx

Evaluate using integration by parts or substitution Check by differentiating n x 9 dx Choose the correct answer below OA x 9 In x 9 C OC x 9 In x 9 x x 9 O E x 9 In x 9 x C C B x 9 In x 9 x x 9 D x In x 9 C OF xln x 9 x C C

Evaluate Assume x 0 Check by differentiating 22x in x dx In 22x in x dx 22 1 1 In x x x 3 3 9 C

Na Du La Cu At Evaluate Be sure to check by differentiating 61 7 1 at dt Determine a change of variables from t to u Choose the correct answer below A u 6t 7 B u t C u 6t 7 D u t 7 Write the integral in terms of u S 6t 7 1 dt S du Type an exact answer Use parentheses to clearly denote the argument of each function

Evaluate the integral using integration by parts Sex5 exd 16x5 exdx

Find the indefinite integral 6u 12u du

Problem Evaluate the following dx Solution Apply integration by parts as follows Now subtract like terms 1 X du 0 1 Contradiction What happened x dx dv dx Sudv uv vdu S dx x x 7 7 dx Sdx 1 f1dx V X Nx 1 Ndx 0 1

3 Find the integral 92 32 7 22 option x A2 3 75 2 3 32 75 3 3 3 7 2 C 312x D 3 7 2 C

Integrate by parts Assome x O x In xdx 11 x

Find the differential of each function an 7 a y tan dy b y 7 2 7t 2 v 2 v dy sec 2 7t 8v 2 1 2

Solve for t to two decimal places 8 0 03t to Do not round until the final answer Then round to the nearest hundredth as neede

The graph of f is given below Sketch a possible graph of f 10 8 10 8 64 2 Clear All Draw 6 4 2 2 4 6 8 101 2 4 6 8 16 ANSICS

Evaluate ff FdS where F is the vector field F x y z rx y zz is the surface of a unit sphere centered at x y z 2 3 1 defined as r 2 y 3 2 1 1 and dS dS points inwards into the surface

Evaluate the integral 16t i 15t2j 10t4k dt

Find the value of the derivative if it exists of the function f x CO 14 2 4 6 8 X x x 16 at the extremum point 0 0

the indefinite integral Remember to use absolute values where appropriate x x x 1 dx 3

Tutorial Exercise Find the general solution of the differential equation and check the result by differentiation dy 16t7 dt Step 1 When solving a differential equation dy 16t7 it is convenient to write it in the equivalent d dt dy 16t7 dt To find the general solution we integrate integrate both sides Thus dy dy y 16 16t7 Step 2 Integrate both sides and use the power rule on the right side 16t7 C dt

Find the indefinite integral and check your result by differentiation Use C for the constant of integration 5 x dx

for what values of x is the series convergent 00 For what values of x is the series n x convergent

Complete the table Use C for the constant of integration Original Integral Rewrite x dx x X dx Integrate C Simplify

Complete the table to find the indefinite integral Use C for the constant of integration Original Integral Rewrite Integrate Simplify 2 dx dx 9 C

O Find the integral 0 2x dx OA 7e 0 2x C OB 1 4e 0 2x C 7e 0 2x 1 0 2x 1 D 35e 0 2x C O c C

0 1 0 1 0 1 0 1 0 1 Evaluate the indefinite integral S 6x 3 dx OA 12x 18x 9x C 72x 36 C B O C 12x 36x C 2 D 36x 36x 36x C

0 1 0 1 0 1 0 1 0 1 Evaluate 1 28x dx O A OB 28 3 OD 28 3 0 56 1 Oc 53 2 C 1 4x 2 C x1 2 C 8x3 2 C

If R is the total resistance of three resistors connected in parallel with resistances R R R3 then 1 1 1 R R 1 1 R If the resistances are measured in ohms as R 82 R 202 and R3 100 2 with a possible error of 0 5 in each case estimate the maximum error in the calculated value of R Roun answer to three decimal places

1 Find the partial fraction decomposition of these rational functions a b c 3x 5 x 3 x 9 d 9 x 1 x 8 7x 9 2x x 2x 3x 1

1 point The graph of the function f x is the horizontal axis is x Given the differential equation a t f x t List the constant or equilibrium solutions to this differential equation in increasing order and indicate whether or not these equilibria are stable semi s unstable

Find the exact value of the distance between the points 3 1 2 and 4 2 3 Let a 1 3 4 3 2 2 8 Find the exact values of 2a 6 a b

ind the area under the graph of f over the interval 2 4 f x he area is 8x 3 5 42 x for x 3 for x 3 Type an integer or a simplified fraction

Maximize Q xy where x and y are positive numbers such that x 6 x 6y 2 The maximum value of Q is and occurs at x and y Type exact answers in simplified form