# Indefinite Integration Questions and Answers

Calculus
Indefinite Integration
Evaluate the integral by making the given substitution Jsec 4 4x tan 4x dx U 4x
Calculus
Indefinite Integration
Consider the indefinite integral U This can be transformed into a basic integral by letting du 3 f and da x 7 dx Performing the substitution yields the integral
Calculus
Indefinite Integration
Evaluate the integral 5 32 fa 26 11 de by making the substitution u x x6 C 11 NOTE Your answer should be in terms of r and not u
Calculus
Indefinite Integration
Given X f x So 0 At what value of a does the local max of f x occur t 49 1 cos t dt
Calculus
Indefinite Integration
For the following function f find the antiderivative F that satisfies the given condition f t sec t F 0 1 F t
Calculus
Indefinite Integration
Determine the following indefinite integral 3 2 24 1 dt S 3 1 214 dt
Calculus
Indefinite Integration
Express the limit lim 7 80 Provide a b and f z in the expression a 2 z 3 z Az over 3 7 as an integral 7 1 1 b a f x dr f x
Calculus
Indefinite Integration
Consider the function f x 42 3x 5x 7 An antiderivative of f z is F z Az Bz C Dz where A is and 13 is and C is and Dis
Calculus
Indefinite Integration
a 4 c 12 5 The co efficient of x in the expansion of 1 2x 3 x 1 a 56 c 154 b 3 d 6 x 1 3 is 6 In the expansion of 71 3 111 9 6561 the number of terms free from radicals is 63 b 65 d 62 a 730 b 729 c 725 d 750 7 The term independent of x in the expansion of 18 b 0 d 3 he coefficients of the 1 15 16 17
Calculus
Indefinite Integration
Next question Let f z 82 z f z 20 z Get a similar question You 8 2 z Eden
Calculus
Indefinite Integration
Write each equation in logarithmic form 3 C
Calculus
Indefinite Integration
A B Integrating both sides indefinitely and using the fact that the integral of a sum is the sum of the integrals we find that D E d sin x de zcos 2 dez sin x dr In this last equation evaluate the indefinite integral on the left side as well as the rightmost indefinite integral on the right Upload picture here ii In the most recent equation from i solve the equation for the expression x cos x dx Upload picture here iii For which product of basic functions have you now found th F
Calculus
Indefinite Integration
2 3 4 5 6 7 8 19 20 21 A E b For each of the following functions use your work in a to help you determine the general antiderivative of the function Recall that the general antiderivative of a function includes C to reflect the entire family of functions that share the same derivative Label each antiderivative by name e g the antiderivative of m should be called M In addition check your work by computing the derivative of each proposed antidarivative B i m x e x ii n x cos 5x 1 D iv v x 2 7x v w x 34 11x c Based on your experience in parts a and b conjecture an antiderivative for each of the following functions Test your conjectures by computing the derivative of each proposed antiderivative i a x cos x ii b x 4x 7 F
Calculus
Indefinite Integration
Find the following indefinite integral antiderivative Ins z Z dz
Calculus
Indefinite Integration
Calculate the following integral We let u So du 1 8 x 8 7x 6 In 3x and then use the integration by parts formula to find that 2 In 32 dr 1 8 In 3x x 8 S and du Hint In 3x and v and finally we finish evaluating to get that x In 3x x x 7x 6 x In 3x dx a In x In 3x dx 1 8 In 3x x 8 1 64 x 8 C
Calculus
Indefinite Integration
Find a polynomial f x of degree 4 that has the following zeros 9 1 4 0 Leave your answer in factored form f x 0 4
Calculus
Indefinite Integration
14 If sin A sin B 3 and angles A and B are acute angles what is the value of cos A B 13 a 12 b d 33 63
Calculus
Indefinite Integration
min f x XERd n n i 1 bi a x Fx X x log 1 ebila
Calculus
Indefinite Integration
Match the following items On the unit circle with its center at 0 0 an arc with central point P 1 0 and terminal point T h length x
Calculus
Indefinite Integration
ffe e 6 1 2 x 2cxy y b k and c are constants
Calculus
Indefinite Integration
When solving using the method of separation of variables the 1D wave equation 0 x t 0 u z 0 0 u 0 t 0 u x t 0 we assume the solution to the Partial Differential Equation and after substitution we get the two Ordinary Differential Equations O D E s A O D E in space where A is the separation constant The C UE subject to u a 0 2 x 2 P D E as u x t A O D E in time and solution to the problem is therefore u z t A cos A Cos B sint 1 1 and B where A Hint Use pi fort and for the fractional use abc d
Calculus
Indefinite Integration
Let f 0 R be a bounded function Consider the functions g h 0 00 h x inf f y y r for every x 0 a 10 points Explain why g h are well defined R that are defined by g x sup f y y a and b 20 points Prove that lim f x 818 lim g x and lim h r exists and HIX 818 exists if and only if the limits lim g x lim h r 348 818
Calculus
Indefinite Integration
3 16 points Find the volume of revolution resulting from the region bounded by x y 2 and y x being revolved around the x axis Use the shell 3x y 2 0 method
Calculus
Indefinite Integration
Question 2 Consider the following Initial Value Problem 4y 20y 24y 48x 48x 48x 92 y 0 7 3 0 5 and let f x be the solution Find f 1
Calculus
Indefinite Integration
7 Find the value of the sum k A 30 B 25 C 576 D 250
Calculus
Indefinite Integration
Points 0 of 1 A ramp is to be built beside the steps to the campus library Find the angle of elevation of the 22 foot ramp to the nearest tenth of a degree if its final height is 7 feet The ramp makes an angle of approximately with the ground Simplify your answer Round to the nearest tenth as needed Save
Calculus
Indefinite Integration
8 points Find the following indefinite integral antiderivative esa 1 6 e5r dx
Calculus
Indefinite Integration
ind the most general antiderivative of the function Check your answer by differentiation Use C for the constant of the antiderivative f a 5 2e0 4q
Calculus
Indefinite Integration
Element X is a radioactive isotope such that every 12 years its mass decreases by half Given that the initial mass of a sample of Element X is 900 grams how much of the element would remain after 6 years to the nearest whole number
Calculus
Indefinite Integration
3 Evaluate tan xdx a c tan x 3 tan x 3 tan x x C tan x C b d tan x 2 tan5x 5 tan x x C C
Calculus
Indefinite Integration
10 Find the area of the region enclose by the curves y x 2 y x x 0 and x 1 31 a 3 32 c H 3 b d 5 65 3
Calculus
Indefinite Integration
9 Find sinxdx a c xcos x 1 x C xsin x 1 x C b d xsin x 1 x C 1x 1 xsin x ln 1 x C
Calculus
Indefinite Integration
13 Evaluate sin 5x cos 4x dx a c co cos x cos 9x C 18 cosx co 18 cos 9x C b d cos 5x c cos 4x C cosx to 1 18 cos 9x C
Calculus
Indefinite Integration
5 38 Find the sum of the geometric series 5 3 A 15 2 B 15 C does not exist 10 5 5 9 27
Calculus
Indefinite Integration
ind the indefinite integral Remember to use absolute values where appropriate Use C for the constant of integra sec x tan x sec x 3 dx
Calculus
Indefinite Integration
Which is an antiderivative of tan²xcscx? secx-1 -cscx tanx+5 No antiderivative exists
Calculus
Indefinite Integration
Let f(x) = 1 + x2 for x E (-0,00) and let g(x)=tan(a) for x E (-1/2, T/2). Which of the following statement is truc? for a E (-1/2, 1/2). d (b)(f(x) + g(x)) = 2x + scc(e) for x € (-7/2, 7/2). (a) e 2x tan(x) d g(x) da f(x)(1+x2)2 = d (c) (f(r)g(x)) = scc²(x)(1+z²) + 2x tan(x) for x = (-1/2, 1/2). dr d (d) f(g(x)) = 2g(x) for r € (-1/2, 1/2). dr (e) None of the above statements is truc.
Calculus
Indefinite Integration
4. EVALUATE THE INTEGRALS. SHOW WORK S 2x + 11 x² + 4x + 5 dx
Calculus
Indefinite Integration
5. EVALUATE THE INTEGRALS. SHOW WORK f x'sec²(3x) dx S x² e5x dx
Calculus
Indefinite Integration
*(d) r³ cos(x²)dr Hint: Rewrite the integrand as r²r cos(x²). Now experiment as to what to set u and du equal to. Do not use online resources!
Calculus
Indefinite Integration
Use Newton's method to find the absolute maximum value of the function f(x) = 2x sin(x), 0 ≤ x ≤ correct to six decimal places.
Calculus
Indefinite Integration
Find f. f(t) = f'(t) sec(t) (sec(t) + tan(t)), = -< < (4) --> -7
Calculus
Indefinite Integration
Consider the following function. x² f(x) = x²-49 (a) Find the critical numbers of f. (Enter your answers as a comma-separated list.) (0,0) x X = (b) Find the open intervals on which the function is increasing or decreasing. (Enter your answers using interval notation. If an answer does not exist, enter DNE.) (-∞, -7)U(-7,0) (0,7) U (7,0) increasing decreasing (c) Apply the First Derivative Test to identify the relative extremum. (If an answer does not exist, enter DNE.) relative maximum (x, y) = DNE x ) relative minimum (x, y) = DNE
Calculus
Indefinite Integration
Find x where ≤x≤ 2T. 2 sec x cos²x = csc²x - cot²x 2+
Calculus
Indefinite Integration
In Problems 35-38, find solutions to the given initial value problem. 35. y" + 3ty' - 6y = 1; y(0) = 0, y'(0) = 0
Calculus
Indefinite Integration
Explain why Rolle's Theorem does not apply to the function even though there exist a and b such that f(a) = f(b). (Select all that apply.) f(x) = [л, 3x] = cot , f(a) does not equal f(b) for all possible values of a and b in the interval [, 3x]. There are points on the interval (a, b) where f is not differentiable. None of these. There are points on the interval [a, b] where f is not continuous. f '(a) does not equal f '(b) for any values in the interval [x, 3x]. X
Calculus
Indefinite Integration
Find the Fourier series of sin²x and cos²x without directly calculating the Fourier coefficients. (Use some standard trigonometric identities)
Calculus
Indefinite Integration
Use Stokes' Theorem to evaluate ∫ ∫S curlF.ds, where F(x, y, z) = (tan-¹(x²yz²), x²y, x² z²) and S is the cone x = √y² + z²,0 ≤ x ≤ 2, oriented in the direction of the positive x-axis. Hint: The boundary of the cone is the circle y² + z² = 4 in x = 2.
Calculus
Indefinite Integration
11:41 "He should do his homework a little better, because he'll find out that we've had a 10 decline of 5 million in the farm population under these government programs. He'll also find that the Democratic administration has sought to get from Congress [an] extension of the farm program to include that three-fourths that is now free." A. Reagan bases his argument on fact - "He should do his homework a little better." B. Reagan bases his argument on fact -- "we've had a decline of 5 million in the farm population." C. Reagan's argument is speculation - "we've had a decline of 5 million in the farm population." D. Reagan bases his argument on emotion -- "we've had a decline of 5 million in the farm population."
Calculus
Indefinite Integration
Let f (x) = -5. #1) Evaluate f(4). f(4)= #2) Solve x for f(x) = -1.