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Breadcrumbs
1. $\displaystyle \int du = u + C$
2. $\displaystyle \int a \, du = a\int du$
3. $\displaystyle \int (du + dv + ... + dz) = \int du + \int dv + ... + \int dz$
4. $\displaystyle \int f (x)\,dx = F(x) + C$
5. $\displaystyle \int_a^b f(x) \, dx = F(b) - F(a)$
6. $\displaystyle \int_a^b f(x) \, dx = -\int_b^a f(x) \, dx$
7. $\displaystyle \int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx$
8. $\displaystyle \int_a^b f(x) \, dx = \int_a^b f(z) \, dz$
9. $\displaystyle \int u^n \, du = \dfrac{u^{n + 1}}{n + 1} + C; \, n \neq -1$
10. $\displaystyle \int \dfrac{du}{u} = \ln u + C$
11. $\displaystyle \int a^u \, du = \dfrac{a^u}{\ln a} + C, \,\, a > 0, \,\, a \neq 1$
12. $\displaystyle \int e^u \, du = e^u + C$
13. $\displaystyle \int \sin u \, du = -\cos u + C$
14. $\displaystyle \int \cos u \, du = \sin u + C$
15. $\displaystyle \int \sec^2 u \, du = \tan u + C$
16. $\displaystyle \int \csc^2 u \, du = -\cot u + C$
17. $\displaystyle \int \sec u \, \tan u \, du = \sec u + C$
18. $\displaystyle \int \csc u \, \cot u \, du = -\csc u + C$
19. $\displaystyle \int \tan u \, du = \ln (\sec u) + C = -\ln (\cos u) + C$
20. $\displaystyle \int \cot u \, du = \ln (\sin u) + C$
21. $\displaystyle \int \sec u \, du = \ln (\sec u + \tan u) + C$
22. $\displaystyle \int \csc u \, du = \ln (\csc u - \cot u) + C = -\ln (\csc u + \cot u) + C$
23. $\displaystyle \int \dfrac{du}{\sqrt{a^2 - u^2}} = \arcsin \, \dfrac{u}{a} + C, \,\,\, a > 0$
24. $\displaystyle \int \dfrac{du}{a^2 + u^2} = \dfrac{1}{a}\arctan \, \dfrac{u}{a} + C$
25. $\displaystyle \int \dfrac{du}{u\sqrt{u^2 - a^2}} = \dfrac{1}{a} {\rm arcsec} \, \dfrac{u}{a} + C$
26. $\displaystyle \int u\,dv = uv - \int v\, du$
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Chapter 30 INTEGRATION OF RATIONAL FUNCTIONS: THE METHOD OF PARTIAL FRACTIONS Chapter 31 INTEGRALS FOR SURFACE AREA, WORK, ... Used thus, 3000 Solved Problems in Calculus can almost serve as a supple-ment to any course in calculus, or even as an independent refresher course. V. This page intentionally left blank . HAPTER 1 nequalities Solve 3 ...
The integral is 1 5 x5 1 4 x4 + 3 x3 + C. Whenever you're working with inde nite inte-grals like this, be sure to write the +C. It signi es that you can add any constant to the antiderivative F(x) to get another one, F(x) + C. When you're working with de nite integrals with limits of integration, Z b a, the constant isn't needed
II. Evaluate the following definite integrals. 3 4 4 22 1 1 5 188 8 1. (5 8 5) 4 5 60 3 3 3 x x x dx x x 3 2 9 5 9 2 2 1 1 2 1026 22 1001 2. ( 2 3) 3 200.2 5 5 5 5 x x x dx x x 9 9 31 22 4 4 1 2 2 20 40 3. ( ) 20 13.333 3 3 3 3 3 x dx x x x 4 32 1 5 5 5 5 75 4. 2.344 2 32 2 32 dx xx 2 34 2 2 1 1 3 44 5 57 5. (1 3 ) 14.25 3 4 3 12 4 tt t t dt 1 ...
CLP-2 Integral Calculus. Joel Feldman University of British Columbia Andrew Rechnitzer University of British Columbia Elyse Yeager University of British Columbia August 23, 2022. iii. CoverDesign: NickLoewen—licensedundertheCC-BY-NC-SA4.0License. Source files: A link to the source files for this document can be found at theCLP textbookwebsite.
Exercises and Problems in Calculus John M. Erdman Portland State University Version August 1, 2013 c 2010 John M. Erdman E-mail address: [email protected]. Contents Preface ix ... MULTIPLE INTEGRALS 267 Chapter 33. DOUBLE INTEGRALS269 33.1. Background269 33.2. Exercises 270 33.3. Problems 274 33.4. Answers to Odd-Numbered Exercises275 Chapter 34 ...
MATH 221 { 1st SEMESTER CALCULUS LECTURE NOTES VERSION 2.0 (fall 2009) This is a self contained set of lecture notes for Math 221. The notes were written by Sigurd Angenent, starting from an extensive collection of notes and problems compiled by Joel Robbin. The LATEX and Python les
Contents Preface xvii 1 Areas, volumes and simple sums 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Areas of simple shapes ...
To compute the indefinite integral R R(x)dx, we need to be able to compute integrals of the form Z a (x n ) dx and Z bx+c (x2 + x+ )m dx: Those of the first type above are simple; a substitution u= x will serve to finish the job. Those of the second type can, via completing the square, be reduced to integrals of the form bx+c (x 2+a)m dx.
Guidelines for Integration by Substitution. 1. Let u be a function of x (usually part of the integrand). 2. Solve for x and dx in terms of u and du. 3. Convert the entire integral to u-variable form and try to fit it to one or more of the basic integration formulas. If none fits, try a different substitution.
calculus is to offer a better way. The problem of integration is to find a limit of sums. The key is to work backward from a limit of differences (which is the derivative). We can integrate v.x/if it turns up as the derivative of another function f.x/. The integral of vDcos xis fDsin x. The integralof vDxis fD 1 2 x2. Basically, f.x/is an ...
MIT18_01SCF10_ex95sol.pdf. pdf. 124 kB. MIT18_01SCF10_ex97sol.pdf. pdf. 93 kB. MIT18_01SCF10_ex98sol.pdf. MIT OpenCourseWare is a web based publication of virtually all MIT course content. OCW is open and available to the world and is a permanent MIT activity.
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Evaluate the integrals in Problems 1—100. The students really should work most of these problems over a period of several days, even while you ... We solve this equation for the desired integral and find that secn-2x tan x n —2 sec" —2 x d". see x dx = This is the desired reduction formula. For example, if we take n =
Fundamental theorem of calculus 1, 2a, 3a, 5a 3E Change of variables; Estimating integrals 6b, 6c Use Applications of Integration (PDF) to do ... Exercises 4J Other applications 2 Solutions. Solutions to Integration problems (PDF) Solutions to Applications of Integration problems (PDF) This problem set is from exercises and solutions written by ...
CHAPTER 0 Highlights of Calculus 0.1 Distance and Speed ==Height and Slope 1 0.2 The Changing Slope of yDx2 and yDxn 9 0.3 The Exponential yDex 15 0.4 Video Summaries and Practice Problems 23 0.5 Graphs and Graphing Calculators 45 CHAPTER 1 Introduction to Calculus 1.1 Velocity and Distance 51 1.2 Calculus Without Limits 59 1.3 The Velocity at ...
Chapter 1 - Fundamental Theorems of Calculus; Chapter 2 - Fundamental Integration Formulas; Chapter 3 - Techniques of Integration; Chapter 4 - Applications of Integration; Book traversal links for Integral Calculus. Chapter 1 - Fundamental Theorems of Calculus
The double integral jjf(x, y)dy dx will now be reduced to single integrals in y and then x. (Or vice versa. Our first integral could equally well be jf(x, y)dx.) Chapter 8 described the same idea for solids of revolution. First came the area of a slice, which is a single integral. Then came a second integral to add up the slices. For solids
4B-7 Solving for 2x 2in y = (x − 1) and y = (x + 1) gives the values a x a -x 2 2 2 top view slice a -x2 2 x = 1 ± √ y and x = −1 ± √ y The hard part is deciding which sign of the square root representing the endpoints of the square.-1 1 1 x = - + y 2 (x- y) 1 x = - y1 Method 1: The point (0, 1) has to be on the two curves.