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Math Calculus EBK SINGLE VARIABLE CALCULUS

Give that ∫ 0 π sin 4 x d x = 3 8 π , what is ∫ π 0 sin 4 θ d θ ?

Give that ∫ 0 π sin 4 x d x = 3 8 π , what is ∫ π 0 sin 4 θ d θ ?

Solution Summary: The author evaluates the integral function using properties of integral.

EBK SINGLE VARIABLE CALCULUS

EBK SINGLE VARIABLE CALCULUS

8th Edition

ISBN: 8220101383693

Author: Stewart

Publisher: Cengage Learning US

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Textbook Question

Chapter 4.2, Problem 42E

Give that ∫ 0 π sin 4 x d x = 3 8 π , what is ∫ π 0 sin 4 θ d θ ?

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Chapter 4.2, Problem 41E

Chapter 4.2, Problem 43E

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Chapter 4 Solutions

EBK SINGLE VARIABLE CALCULUS

Ch. 4.1 - Prob. 1E Ch. 4.1 - (a) Use six rectangles to find estimates of each...Ch. 4.1 - (a) Estimate the area under the graph of f(x) =...Ch. 4.1 - Prob. 4E Ch. 4.1 - (a) Estimate the area under the graph of f(x) = 1...Ch. 4.1 - Prob. 6E Ch. 4.1 - Prob. 7E Ch. 4.1 - Prob. 8E Ch. 4.1 - With a programmable calculator (or a computer), it...Ch. 4.1 - With a programmable calculator (or a computer), it...

Ch. 4.1 - Prob. 11E Ch. 4.1 - Prob. 12E Ch. 4.1 - The speed of a runner increased steadily during...Ch. 4.1 - Prob. 14E Ch. 4.1 - Oil leaked from a tank at a rate of r(t) liters...Ch. 4.1 - Prob. 16E Ch. 4.1 - The velocity graph of a braking car is shown. Use...Ch. 4.1 - The velocity graph of a car accelerating from rest...Ch. 4.1 - In someone infected with measles, the virus level...Ch. 4.1 - The table shows the number of people per day who...Ch. 4.1 - Use Definition 2 to find an expression for the...Ch. 4.1 - Prob. 22E Ch. 4.1 - Use Definition 2 to find an expression for the...Ch. 4.1 - Prob. 24E Ch. 4.1 - Prob. 25E Ch. 4.1 - (a) Use Definition 2 to find an expression for the...Ch. 4.1 - Let A be the area under the graph of an increasing...Ch. 4.1 - If A is the area under the curve y = sin x from 0...Ch. 4.1 - (a) Express the area under the curve y = x5 from 0...Ch. 4.1 - (a) Express the area under the curve y = x4 + 5x2...Ch. 4.1 - Prob. 31E Ch. 4.1 - (a) Let An be the area of a polygon with n equal...Ch. 4.2 - Evaluate the Riemann sum for f(x)=x1,6x4, with...Ch. 4.2 - If f(x)=cosx0x3/4 evaluate the Riemann sum with n...Ch. 4.2 - If f(x)=x24,0x3, find the Riemann sum with n = 6,...Ch. 4.2 - (a) Find the Riemann sum for f(x)=1/x,1x2, with...Ch. 4.2 - The graph of a function f is given. Estimate...Ch. 4.2 - The graph of g is shown. Estimate 24g(x)dx with...Ch. 4.2 - A table of values of an increasing function f is...Ch. 4.2 - Prob. 8E Ch. 4.2 - Prob. 9E Ch. 4.2 - Prob. 10E Ch. 4.2 - Prob. 11E Ch. 4.2 - Prob. 12E Ch. 4.2 - If you have a CAS that evaluates midpoint...Ch. 4.2 - Prob. 14E Ch. 4.2 - Prob. 15E Ch. 4.2 - Prob. 16E Ch. 4.2 - Express the limit as a definite integral on the...Ch. 4.2 - Express the limit as a definite integral on the...Ch. 4.2 - Express the limit as a definite integral on the...Ch. 4.2 - Prob. 20E Ch. 4.2 - Use the form of the definition of the integral...Ch. 4.2 - Use the form of the definition of the integral...Ch. 4.2 - Use the form of the definition of the integral...Ch. 4.2 - Use the form of the definition of the integral...Ch. 4.2 - Use the form of the definition of the integral...Ch. 4.2 - Prob. 26E Ch. 4.2 - Prove that abxdx=b2a22.Ch. 4.2 - Prob. 28E Ch. 4.2 - Prob. 29E Ch. 4.2 - Prob. 30E Ch. 4.2 - Express the integral as a limit of sums. Then...Ch. 4.2 - Prob. 32E Ch. 4.2 - The graph of f is shown. Evaluate each integral by...Ch. 4.2 - The graph of g consists of two straight lines and...Ch. 4.2 - Evaluate the integral by interpreting it in terms...Ch. 4.2 - Evaluate the integral by interpreting it in terms...Ch. 4.2 - Evaluate the integral by interpreting it in terms...Ch. 4.2 - Evaluate the integral by interpreting it in terms...Ch. 4.2 - Prob. 39E Ch. 4.2 - Evaluate the integral by interpreting it in terms...Ch. 4.2 - Prob. 41E Ch. 4.2 - Give that 0sin4xdx=38, what is 0sin4d?Ch. 4.2 - In Example 4.1.2 we showed that 01x2dx=13. Use...Ch. 4.2 - Prob. 44E Ch. 4.2 - Prob. 45E Ch. 4.2 - Prob. 46E Ch. 4.2 - Prob. 47E Ch. 4.2 - If 28f(x)dx=7.3 and 24f(x)dx=5.9, find 48f(x)dx.Ch. 4.2 - Prob. 49E Ch. 4.2 - Prob. 50E Ch. 4.2 - Prob. 51E Ch. 4.2 - Prob. 52E Ch. 4.2 - Prob. 53E Ch. 4.2 - Suppose f has absolute minimum value m and...Ch. 4.2 - Prob. 55E Ch. 4.2 - Prob. 56E Ch. 4.2 - Use the properties of integrals to verify the...Ch. 4.2 - Prob. 58E Ch. 4.2 - Prob. 59E Ch. 4.2 - Prob. 60E Ch. 4.2 - Prob. 61E Ch. 4.2 - Prob. 62E Ch. 4.2 - Prob. 63E Ch. 4.2 - Prob. 64E Ch. 4.2 - Prob. 65E Ch. 4.2 - Prob. 66E Ch. 4.2 - Prob. 67E Ch. 4.2 - Prob. 68E Ch. 4.2 - Prob. 69E Ch. 4.2 - Prob. 70E Ch. 4.2 - Prob. 71E Ch. 4.2 - Prob. 72E Ch. 4.2 - Express the limit as a definite integral. 73....Ch. 4.2 - Prob. 74E Ch. 4.2 - Prob. 75E Ch. 4.3 - Explain exactly what is meant by the statement...Ch. 4.3 - Prob. 2E Ch. 4.3 - Let g(x)=0xf(t)dt, where f is the function whose...Ch. 4.3 - Prob. 4E Ch. 4.3 - Sketch the area represented by g(x). Then find...Ch. 4.3 - Prob. 6E Ch. 4.3 - Use Part 1 of the Fundamental Theorem of Calculus...Ch. 4.3 - Prob. 8E Ch. 4.3 - Use Part 1 of the Fundamental Theorem of Calculus...Ch. 4.3 - Prob. 10E Ch. 4.3 - Use Part 1 of the Fundamental Theorem of Calculus...Ch. 4.3 - Prob. 12E Ch. 4.3 - Use Part 1 of the Fundamental Theorem of Calculus...Ch. 4.3 - Use Part 1 of the Fundamental Theorem of Calculus...Ch. 4.3 - Use Part 1 of the Fundamental Theorem of Calculus...Ch. 4.3 - Prob. 16E Ch. 4.3 - Use Part 1 of the Fundamental Theorem of Calculus...Ch. 4.3 - Prob. 18E Ch. 4.3 - Evaluate the integral. 19. 13(x2+2x4)dx Ch. 4.3 - Prob. 20E Ch. 4.3 - Prob. 21E Ch. 4.3 - Prob. 22E Ch. 4.3 - Evaluate the integral. 23. 19xdx Ch. 4.3 - Prob. 24E Ch. 4.3 - Evaluate the integral. 25. /6sind Ch. 4.3 - Prob. 26E Ch. 4.3 - Evaluate the integral. 27. 01(u+2)(u3)du Ch. 4.3 - Prob. 28E Ch. 4.3 - Evaluate the integral. 29. 142+x2xdx Ch. 4.3 - Prob. 30E Ch. 4.3 - Evaluate the integral. 31. /6/2csctcottdt Ch. 4.3 - Prob. 32E Ch. 4.3 - Prob. 33E Ch. 4.3 - Prob. 34E Ch. 4.3 - Evaluate the integral. 35. 12v5+3v6v4dv Ch. 4.3 - Prob. 36E Ch. 4.3 - Prob. 37E Ch. 4.3 - Prob. 38E Ch. 4.3 - Prob. 39E Ch. 4.3 - Prob. 40E Ch. 4.3 - Prob. 41E Ch. 4.3 - Prob. 42E Ch. 4.3 - Use a graph to give a rough estimate of the area...Ch. 4.3 - Prob. 44E Ch. 4.3 - Prob. 45E Ch. 4.3 - Use a graph to give a rough estimate of the area...Ch. 4.3 - Prob. 47E Ch. 4.3 - Prob. 48E Ch. 4.3 - Prob. 49E Ch. 4.3 - Prob. 50E Ch. 4.3 - Prob. 51E Ch. 4.3 - Prob. 52E Ch. 4.3 - Prob. 53E Ch. 4.3 - Prob. 54E Ch. 4.3 - Prob. 55E Ch. 4.3 - Prob. 56E Ch. 4.3 - Prob. 57E Ch. 4.3 - If f(x)=0x(1t2)cos2tdt, on what interval is f...Ch. 4.3 - On what interval is the curve y=0xt2t2+t+2dt...Ch. 4.3 - Prob. 60E Ch. 4.3 - Prob. 61E Ch. 4.3 - Prob. 62E Ch. 4.3 - The Fresnel function S was defined in Example 3...Ch. 4.3 - The sine integral function Si(x)=0xsinttdt is...Ch. 4.3 - Let g(x)=0xf(t)dt, where f is the function whose...Ch. 4.3 - Let g(x)=0xf(t)dt, where f is the function whose...Ch. 4.3 - Prob. 67E Ch. 4.3 - Prob. 68E Ch. 4.3 - Prob. 69E Ch. 4.3 - Prob. 70E Ch. 4.3 - Prob. 71E Ch. 4.3 - (a) Show that cos(x2) cos x for 0 x 1. (b)...Ch. 4.3 - Show that 0510x2x4+x2+1dx0.1 by comparing the...Ch. 4.3 - Let f(x)={0ifx0xif0x12xif1x20ifx2 and...Ch. 4.3 - Find a function f and a number a such that...Ch. 4.3 - Prob. 76E Ch. 4.3 - A manufacturing company owns a major piece of...Ch. 4.3 - A high-tech company purchases a new computing...Ch. 4.3 - Evaluate the integral. 79. 1912xdx Ch. 4.3 - Prob. 80E Ch. 4.3 - Prob. 81E Ch. 4.3 - Prob. 82E Ch. 4.3 - Prob. 83E Ch. 4.3 - Prob. 84E Ch. 4.4 - Verify by differentiation that the formula is...Ch. 4.4 - Verify by differentiation that the formula is...Ch. 4.4 - Prob. 3E Ch. 4.4 - Prob. 4E Ch. 4.4 - Find the general indefinite integral. 5....Ch. 4.4 - Prob. 6E Ch. 4.4 - Find the general indefinite integral. 7....Ch. 4.4 - Prob. 8E Ch. 4.4 - Find the general indefinite integral. 9....Ch. 4.4 - Prob. 10E Ch. 4.4 - Find the general indefinite integral. 11. 1+x+xxdx Ch. 4.4 - Prob. 12E Ch. 4.4 - Find the general indefinite integral. 13....Ch. 4.4 - Prob. 14E Ch. 4.4 - Find the general indefinite integral. 15....Ch. 4.4 - Prob. 16E Ch. 4.4 - Find the general indefinite integral. Illustrate...Ch. 4.4 - Prob. 18E Ch. 4.4 - Evaluate the integral. 19. 23(x23)dx Ch. 4.4 - Prob. 20E Ch. 4.4 - Evaluate the integral. 21. 20(12t4+14t3t)dt Ch. 4.4 - Prob. 22E Ch. 4.4 - Evaluate the integral. 23. 02(2x3)(4x2+1)dx Ch. 4.4 - Prob. 24E Ch. 4.4 - Evaluate the integral. 25. 0(4sin3cos)d Ch. 4.4 - Prob. 26E Ch. 4.4 - Prob. 27E Ch. 4.4 - Prob. 28E Ch. 4.4 - Prob. 29E Ch. 4.4 - Prob. 30E Ch. 4.4 - Prob. 31E Ch. 4.4 - Prob. 32E Ch. 4.4 - Prob. 33E Ch. 4.4 - Prob. 34E Ch. 4.4 - Prob. 35E Ch. 4.4 - Prob. 36E Ch. 4.4 - Prob. 37E Ch. 4.4 - Prob. 38E Ch. 4.4 - Prob. 39E Ch. 4.4 - Prob. 40E Ch. 4.4 - Prob. 41E Ch. 4.4 - Prob. 42E Ch. 4.4 - Prob. 43E Ch. 4.4 - Repeat Exercise 43 for the curve y = 2x + 3x4 ...Ch. 4.4 - The area of the region that lies to the right of...Ch. 4.4 - Prob. 46E Ch. 4.4 - If w'(t) is the rate of growth of a child in...Ch. 4.4 - Prob. 48E Ch. 4.4 - If oil leaks from a tank at a rate of r(t) gallons...Ch. 4.4 - A honeybee population starts with 100 bees and...Ch. 4.4 - In Section 3.7 we defined the marginal revenue...Ch. 4.4 - Prob. 52E Ch. 4.4 - If x is measured in meters and f(x) is measured in...Ch. 4.4 - If the units for x are feet and the units for a(x)...Ch. 4.4 - The velocity function (in meters per second) is...Ch. 4.4 - The velocity function (in meters per second) is...Ch. 4.4 - Prob. 57E Ch. 4.4 - Prob. 58E Ch. 4.4 - The linear density of a rod of length 4 m is given...Ch. 4.4 - Water flows from the bottom of a storage tank at a...Ch. 4.4 - Prob. 61E Ch. 4.4 - Suppose that a volcano is erupting and readings of...Ch. 4.4 - Prob. 63E Ch. 4.4 - Prob. 64E Ch. 4.4 - The graph of the acceleration a(t) of a car...Ch. 4.4 - Shown is the graph of traffic on an Internet...Ch. 4.4 - The following graph shows the power consumption in...Ch. 4.4 - Prob. 68E Ch. 4.4 - Evaluate the integral. 69. (sinx+sinhx)dx Ch. 4.4 - Prob. 70E Ch. 4.4 - Prob. 71E Ch. 4.4 - Prob. 72E Ch. 4.4 - Prob. 73E Ch. 4.4 - The area labeled B is three times the area labeled...Ch. 4.5 - Evaluate the integral by making the given...Ch. 4.5 - Prob. 2E Ch. 4.5 - Evaluate the integral by making the given...Ch. 4.5 - Prob. 4E Ch. 4.5 - Evaluate the integral by making the given...Ch. 4.5 - Prob. 6E Ch. 4.5 - Evaluate the indefinite integral. 7. x1x2dx Ch. 4.5 - Prob. 8E Ch. 4.5 - Evaluate the indefinite integral. 9. (12x)9dx Ch. 4.5 - Prob. 10E Ch. 4.5 - Evaluate the indefinite integral. 11. sin(2/3)d Ch. 4.5 - Prob. 12E Ch. 4.5 - Evaluate the indefinite integral. 13. sec3ttan3tdt Ch. 4.5 - Prob. 14E Ch. 4.5 - Evaluate the indefinite integral. 15. cos(1+5t)dt Ch. 4.5 - Prob. 16E Ch. 4.5 - Evaluate the indefinite integral. 17. sec2tan3d Ch. 4.5 - Prob. 18E Ch. 4.5 - Evaluate the indefinite integral. 19....Ch. 4.5 - Prob. 20E Ch. 4.5 - Prob. 21E Ch. 4.5 - Evaluate the indefinite integral. 22. cos(/x)x2dx Ch. 4.5 - Evaluate the indefinite integral. 23. z21+z33dz Ch. 4.5 - Prob. 24E Ch. 4.5 - Evaluate the indefinite integral. 25. cotxcsc2xdx Ch. 4.5 - Prob. 26E Ch. 4.5 - Evaluate the indefinite integral. 27. sec3xtanxdx Ch. 4.5 - Prob. 28E Ch. 4.5 - Evaluate the indefinite integral. 29. x(2x+5)8dx Ch. 4.5 - Prob. 30E Ch. 4.5 - Prob. 31E Ch. 4.5 - Prob. 32E Ch. 4.5 - Prob. 33E Ch. 4.5 - Prob. 34E Ch. 4.5 - Evaluate the definite integral. 35. 01cos(t/2)dt Ch. 4.5 - Prob. 36E Ch. 4.5 - Evaluate the definite integral. 37. 011+7x3dx Ch. 4.5 - Prob. 38E Ch. 4.5 - Prob. 39E Ch. 4.5 - Prob. 40E Ch. 4.5 - Prob. 41E Ch. 4.5 - Prob. 42E Ch. 4.5 - Evaluate the definite integral. 43. 013dx(1+2x)23 Ch. 4.5 - Prob. 44E Ch. 4.5 - Evaluate the definite integral. 45. 0axx2+a2dx(a0)Ch. 4.5 - Prob. 46E Ch. 4.5 - Evaluate the definite integral. 47. 12xx1dx Ch. 4.5 - Prob. 48E Ch. 4.5 - Evaluate the definite integral. 49....Ch. 4.5 - Prob. 50E Ch. 4.5 - Prob. 51E Ch. 4.5 - Prob. 52E Ch. 4.5 - Use a graph to give a rough estimate of the area...Ch. 4.5 - Use a graph to give a rough estimate of the area...Ch. 4.5 - Prob. 55E Ch. 4.5 - Prob. 56E Ch. 4.5 - Breathing is cyclic and a full respiratory cycle...Ch. 4.5 - A model for the basal metabolism rate, in kcal/h,...Ch. 4.5 - Prob. 59E Ch. 4.5 - Prob. 60E Ch. 4.5 - Prob. 61E Ch. 4.5 - Prob. 62E Ch. 4.5 - If a and b are positive numbers, show that...Ch. 4.5 - If f is continuous on [0, ], use the substitution...Ch. 4.5 - If f is continuous, prove that...Ch. 4.5 - Prob. 66E Ch. 4.5 - Prob. 67E Ch. 4.5 - Prob. 68E Ch. 4.5 - Prob. 69E Ch. 4.5 - Prob. 70E Ch. 4.5 - Prob. 71E Ch. 4.5 - Prob. 72E Ch. 4.5 - Prob. 73E Ch. 4.5 - Prob. 74E Ch. 4.5 - Prob. 75E Ch. 4.5 - Evaluate the integral. 76. sin(lnx)xdx Ch. 4.5 - Prob. 77E Ch. 4.5 - Prob. 78E Ch. 4.5 - Prob. 79E Ch. 4.5 - Prob. 80E Ch. 4.5 - Prob. 81E Ch. 4.5 - Prob. 82E Ch. 4.5 - Evaluate the integral. 83. 01ez+1ez+zdz Ch. 4.5 - Prob. 84E Ch. 4.5 - Prob. 85E Ch. 4 - (a) Write an expression for a Riemann sum of a...Ch. 4 - (a) Write the definition of the definite integral...Ch. 4 - State the Midpoint Rule.Ch. 4 - State both parts of the Fundamental Theorem of...Ch. 4 - (a) State the Net Change Theorem. (b) If r(t) is...Ch. 4 - Suppose a particle moves back and forth along a...Ch. 4 - Prob. 7RCC Ch. 4 - Explain exactly what is meant by the statement...Ch. 4 - Prob. 9RCC Ch. 4 - Prob. 1RQ Ch. 4 - Prob. 2RQ Ch. 4 - Prob. 3RQ Ch. 4 - Prob. 4RQ Ch. 4 - Prob. 5RQ Ch. 4 - Prob. 6RQ Ch. 4 - Prob. 7RQ Ch. 4 - Prob. 8RQ Ch. 4 - Prob. 9RQ Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Prob. 11RQ Ch. 4 - Prob. 12RQ Ch. 4 - Prob. 13RQ Ch. 4 - Prob. 14RQ Ch. 4 - Prob. 15RQ Ch. 4 - Prob. 16RQ Ch. 4 - Prob. 17RQ Ch. 4 - Determine whether the statement is true or false....Ch. 4 - Use the given graph of f to find the Riemann sum...Ch. 4 - Prob. 2RE Ch. 4 - Prob. 3RE Ch. 4 - Prob. 4RE Ch. 4 - Prob. 5RE Ch. 4 - Prob. 6RE Ch. 4 - The figure shows the graphs of f,f, and 0xf(t)dt....Ch. 4 - Evaluate: (a) 0/2ddx(sinx2cosx3)dx (b)...Ch. 4 - Prob. 9RE Ch. 4 - Prob. 10RE Ch. 4 - Prob. 11RE Ch. 4 - Prob. 12RE Ch. 4 - Prob. 13RE Ch. 4 - Prob. 14RE Ch. 4 - Prob. 15RE Ch. 4 - Prob. 16RE Ch. 4 - Prob. 17RE Ch. 4 - Prob. 18RE Ch. 4 - Prob. 19RE Ch. 4 - Prob. 20RE Ch. 4 - Prob. 21RE Ch. 4 - Prob. 22RE Ch. 4 - Prob. 23RE Ch. 4 - Prob. 24RE Ch. 4 - Prob. 25RE Ch. 4 - Prob. 26RE Ch. 4 - Prob. 27RE Ch. 4 - Prob. 28RE Ch. 4 - Prob. 29RE Ch. 4 - Prob. 30RE Ch. 4 - Prob. 31RE Ch. 4 - Prob. 32RE Ch. 4 - Prob. 33RE Ch. 4 - Prob. 34RE Ch. 4 - Prob. 35RE Ch. 4 - Prob. 36RE Ch. 4 - Prob. 37RE Ch. 4 - Prob. 38RE Ch. 4 - Prob. 39RE Ch. 4 - Prob. 40RE Ch. 4 - Prob. 41RE Ch. 4 - Prob. 42RE Ch. 4 - Prob. 43RE Ch. 4 - Prob. 44RE Ch. 4 - Prob. 45RE Ch. 4 - A particle moves along a line with velocity...Ch. 4 - Prob. 47RE Ch. 4 - Prob. 48RE Ch. 4 - A population of honeybees increased at a rate of...Ch. 4 - Prob. 50RE Ch. 4 - If f is continuous and 02f(x)dx=6, evaluate...Ch. 4 - The Fresnel function S(x)=0xsin(12t2)dt was...Ch. 4 - Prob. 53RE Ch. 4 - Prob. 54RE Ch. 4 - Prob. 55RE Ch. 4 - Prob. 56RE Ch. 4 - If f is continuous on [0, 1], prove that...Ch. 4 - Prob. 58RE Ch. 4 - Prob. 1P Ch. 4 - Prob. 2P Ch. 4 - Prob. 3P Ch. 4 - (a) Graph several members of the family of...Ch. 4 - Prob. 5P Ch. 4 - If f(x)=0xx2sin(t2)dt, find f(x).Ch. 4 - Prob. 7P Ch. 4 - Prob. 8P Ch. 4 - Prob. 9P Ch. 4 - Find d2dx20x(1sint1+u4du)dt.Ch. 4 - Suppose the coefficients of the cubic polynomial...Ch. 4 - Prob. 12P Ch. 4 - Prob. 13P Ch. 4 - The figure shows a parabolic segment, that is. a...Ch. 4 - Prob. 15P Ch. 4 - Prob. 16P Ch. 4 - Evaluate limn(1nn+1+1nn+2++1nn+n).Ch. 4 - For any number c, we let fc(x) be the smaller of...

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Fundamental Trigonometric Identities: Reciprocal, Quotient, and Pythagorean Identities; Author: Mathispower4u;https://www.youtube.com/watch?v=OmJ5fxyXrfg;License: Standard YouTube License, CC-BY