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All Textbook Solutions for Calculus: Early Transcendentals
Evaluate the integral. 17. sin2xsin2xdxEvaluate the integral. 18. sinxcos(12x)dxEvaluate the integral. 19. tsin2tdtEvaluate the integral. 20. xsin3xdxEvaluate the integral. 21. tanxsec3xdxEvaluate the integral. 22. tan2sec4dEvaluate the integral. 23. tan2xdxEvaluate the integral. 24. (tan2x+tan4x)dxEvaluate the integral. 25. tan4xsec6xdxEvaluate the integral. 26. 0/4sec6tan6dEvaluate the integral. 27. tan3xsecxdxEvaluate the integral. 28. tan5xsec3xdxEvaluate the integral. 29. tan3xsec6xdxEvaluate the integral. 30. 0/4tan3tdtEvaluate the integral. 31. tan5xdxEvaluate the integral. 32. tan2xsecxdxEvaluate the integral. 33. xsecxtanxdxEvaluate the integral. 34. sincos3dEvaluate the integral. 35. /6/2cot2xdxEvaluate the integral. 36. /4/2cot3xdxEvaluate the integral. 37. /4/2cot5csc3dEvaluate the integral. 38. /4/2csc4cot4dEvaluate the integral. 39. cscxdxEvaluate the integral. 40. /6/3csc3xdxEvaluate the integral. 41. sin8xcos5xdxEvaluate the integral. 42. sin2sin6dEvaluate the integral. 43. 0/2cot5tcos10tdtEvaluate the integral. 44. sinxsec5xdxEvaluate the integral. 45. 0/61+cos2xdxEvaluate the integral. 46. 0/41cos4dEvaluate the integral. 47. 1tan2xsec2xdxEvaluate the integral. 48. dxcosx1Evaluate the integral. 49. xtan2xdxIf 0/4tan6xsecxdx=I, express the value of 0/4tan8xsecxdx in terms of IEvaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the integrand and its antiderivative (taking C = 0). 51. xsin2(x2)dxEvaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the integrand and its antiderivative (taking C = 0). 52. sin5cos3xdxEvaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the integrand and its antiderivative (taking C = 0). 53. sin3xsin6xdxEvaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the integrand and its antiderivative (taking C = 0). 54. sec4(12x)dxFind the average value of the function f(x) = sin2x cos3x on the interval [ , ].Evaluate sin x cos x dx by four methods: (a) the substitution u = cos x (b) the substitution u = sin x (c) the identity sin 2x = 2 sin x cos x (d) integration by parts Explain the different appearances of the answers.Find the area of the region bounded by the given curves. 57. y = sin2x, y = sin3x, 0 xFind the area of the region bounded by the given curves. 58. y = tan x, y = tan2x, 0 x /4Use a graph of the integrand to guess the value of the integral. Then use the methods of this section to prove that your guess is correct. 59. 02cos3xdxUse a graph of the integrand to guess the value of the integral. Then use the methods of this section to prove that your guess is correct. 60. 02sin2xcos5xdxFind the volume obtained by rotating the region bounded by the curves about the given axis. 61. y = sin x, y = 0, /2 x ; about the x-axisFind the volume obtained by rotating the region bounded by the curves about the given axis. 62. y = sin2 x, y = 0, 0 x ; about the x-axisFind the volume obtained by rotating the region bounded by the curves about the given axis. 63. y = sin x, y = cos x, 0 x /4; about y = 1Find the volume obtained by rotating the region bounded by the curves about the given axis. 64. y = sec x, y = cos x, 0 x /3; about y = 1A particle moves on a straight line with velocity function (t) = sin t cos2 t. Find its position function s = f(t) if f(0) = 0.Household electricity is supplied in the form of alternating current that varies from 155 V to 155 V with a frequency of 60 cycles per second (Hz). The voltage is thus given by the equation E(t)=155sin(120t) where t is the time in seconds. Voltmeters read the RMS (root-mean-square) voltage, which is the square root of the average value of [E(t)]2 over one cycle. (a) Calculate the RMS voltage of household current. (b) Many electric stoves require an RMS voltage of 220 V. Find the corresponding amplitude A needed for the voltage E(t) = A sin(l20t).Prove the formula, where m and n are positive integers. 67. sinmxcosnxdx=0Prove the formula, where m and n are positive integers. 68. sinmxsinnxdx={0ifmnifm=nProve the formula, where m and n are positive integers. 69. cosmxcosnxdx={0ifmnifm=nA finite Fourier series is given by the sum f(x)=n=1Nansinnx=a1sinx+a2sin2x++aNsinNx Show that the mth coefficient am is given by the formula am=1=f(x)sinmxdxEvaluate the integral using the indicated trigonometric substitution. Sketch and label the associated right triangle. 1. dxx24x2 x =2 sinEvaluate the integral using the indicated trigonometric substitution. Sketch and label the associated right triangle. 2. x3x2+4dx x =2 tanEvaluate the integral using the indicated trigonometric substitution. Sketch and label the associated right triangle. 3. x24xdx x =2 secEvaluate the integral. 4. x29x2dxEvaluate the integral. 5. x21x4dxEvaluate the integral. 6. 03x36x2dxEvaluate the integral. 7. 0adx(a2+x2)3/2, a 0Evaluate the integral. 8. dtt2t216Evaluate the integral. 9. 23dx(x21)3/2Evaluate the integral. 10. 02/349x2dxEvaluate the integral. 11. 01/2x14x2dxEvaluate the integral. 12. 02dt4+t2Evaluate the integral. 13. x29x3dxEvaluate the integral. 14. 01dx(x2+1)2Evaluate the integral. 15. 0ax2a2x2dxEvaluate the integral. 16. 2/32/3dxx59x21Evaluate the integral. 17. xx27dxEvaluate the integral. 18. dx[(ax)2b2]3/2Evaluate the integral. 19. 1+x2xdxEvaluate the integral. 20.x1+x2dxEvaluate the integral. 21.00.6x2925x2dxEvaluate the integral. 22. 01x2+1dxEvaluate the integral. 23. dxx2+2x+5Evaluate the integral. 24. 01xx2dxEvaluate the integral. 25. x23+2xx2dxEvaluate the integral. 26. x2(3+4x4x2)3/2dxEvaluate the integral. 27. x2+2xdxEvaluate the integral. 28. x2+1(x22x+2)2dxEvaluate the integral. 29. x1x4dxEvaluate the integral. 30. 0/2cost1+sin2tdt(a) Use trigonometric substitution to show that dxx2+a2=ln(x+x2+a2)+C (b) Use the hyperbolic substitution x = a sinh t to show that dxx2+a2=sinh1(xa)+C These formulas are connected by Formula 3.11.3.Evaluate x2(x2+a2)3/2dx (a) by trigonometric substitution. (b) by the hyperbolic substitution x = a sinh t.Find the average value of f(x)=x21/x, 1 x 1.Find the area of the region bounded by the hyperbola 9x2 - 4y2 = 36 and the line x = 3.Prove the formula A = 12r2 for the area of a sector of a circle with radius r and central angle . [Hint: Assume 0 7 /2 and place the center of the circle at the origin so it has the equation x2 + y2 = r2. Then A is the sum of the area of the triangle POQ and the area of the region PQR in the figure.]Evaluate the integral dxx4x22 Graph the integrand and its indefinite integral on the same screen and check that your answer is reasonable.Find the volume of the solid obtained by rotating about the x-axis the region enclosed by the curves y = 9/(x2 + 9). y = 0, x = 0, and x = 3.Find the volume of the solid obtained by rotating about the line x = 1 the region under the curve y = x1x2, 0 x 1.(a) Use trigonometric substitution to verify that 0xa2t2dt=12a2sin1(x/a)+12xa2x2 (b) Use the figure to give trigonometric interpretations of both terms on the right side of the equation in part (a).The parabola y = 12x2 divides the disk x2 + y2 8 into two parts. Find the areas of both parts.A torus is generated by rotating the circle x2 + (y R)2 = r2 about the x-axis. Find the volume enclosed by the torus.A charged rod of length L produces an electric field at point P(a, b) given by E(P)=aLab40(x2+b2)3/2dx where is the charge density per unit length on the rod and 0 is the free space permittivity (see the figure). Evaluate the integral to determine an expression for the electric field E(P).Find the area of the crescent-shaped region (called a lune) bounded by arcs of circles with radii r and R. (See the figure.)A water storage tank has the shape of a cylinder with diameter 10 ft. It is mounted so that the circular cross-sections are vertical. If the depth of the water is 7 ft, what percentage of the total capacity is being used?Write out the form of the partial fraction decomposition of the function (as in Example 7). Do not determine the numerical values of the coefficients. 1. (a) 4+x(1+2x)(3x) (b) 1xx3+x4Write out the form of the partial fraction decomposition of the function (as in Example 7). Do not determine the numerical values of the coefficients. 2. (a) x6x2+x6 (b) x2x2+x+6Write out the form of the partial fraction decomposition of the function (as in Example 7). Do not determine the numerical values of the coefficients. 3. (a) 1x2+x4 (b) x3+1x33x2+2xWrite out the form of the partial fraction decomposition of the function (as in Example 7). Do not determine the numerical values of the coefficients. 4. (a) x42x3+x2+2x1x22x+1 (b) x21x3+x2+xWrite out the form of the partial fraction decomposition of the function (as in Example 7). Do not determine the numerical values of the coefficients. 5. (a) x6x24 (b) x4(x2x+1)(x2+2)2Write out the form of the partial fraction decomposition of the function (as in Example 7). Do not determine the numerical values of the coefficients. 6. (a) t6+1t6+t3 (b) x5+1(x2x)(x4+2x2+1)Evaluate the integral. 7.x4x1dxEvaluate the integral. 8.3t2t+1dtEvaluate the integral. 9.5x+1(2x+1)(x1)dxEvaluate the integral. 10.y(y+4)(2y1)dyEvaluate the integral. 11.0122x2+3x+1dxEvaluate the integral. 12.01x4x25x+6dxEvaluate the integral. 13.axx2bxdxEvaluate the integral. 14.1(x+a)(x+b)dxEvaluate the integral. 15.10x34x+1x23x+2dxEvaluate the integral. 16.12x3+4x2+x1x3+x2dxEvaluate the integral. 17.124y27y12y(y+2)(y3)dyEvaluate the integral. 18.123x2+6x+2x2+3x+2dxEvaluate the integral. 19.01x2+x+1(x+1)2(x+2)dxEvaluate the integral. 20.23x(35x)(3x1)(x1)2dxEvaluate the integral. 21.dt(t21)2Evaluate the integral. 22.x4+9x2+x+2x2+9dxEvaluate the integral. 23.10(x1)(x2+9)dxEvaluate the integral. 24.x2x+6x3+3xdxEvaluate the integral. 25.4xx3+x2+x+1dxEvaluate the integral. 26.x2+x+1(x2+1)2dxEvaluate the integral. 27.x3+4x+3x4+5x2+4dxEvaluate the integral. 28.x3+6x2x4+6x2dxEvaluate the integral. 29.x+4x2+2x+5dxEvaluate the integral. 30.x32x2+2x5x4+4x2+3dxEvaluate the integral. 31.1x31dxEvaluate the integral. 32.01xx2+4x+13dxEvaluate the integral. 33.01x3+2xx4+4x2+3dxEvaluate the integral. 34.x5+x1x3+1dxEvaluate the integral. 35.5x4+7x2+x+2x(x2+1)2dxEvaluate the integral. 36.x4+3x2+1x5+5x3+5xdxEvaluate the integral. 37.x23x+7(x24x+6)2dxEvaluate the integral. 38.x3+2x2+3x2(x2+2x+2)2dxMake a substitution to express the integrand as a rational function and then evaluate the integral. 39.dxxx1Make a substitution to express the integrand as a rational function and then evaluate the integral. 40.dx2x+3+xMake a substitution to express the integrand as a rational function and then evaluate the integral. 41.dxx2+xxMake a substitution to express the integrand as a rational function and then evaluate the integral. 42.0111+x3dxMake a substitution to express the integrand as a rational function and then evaluate the integral. 43.x3x2+13dxMake a substitution to express the integrand as a rational function and then evaluate the integral. 44.dx(1+x)2Make a substitution to express the integrand as a rational function and then evaluate the integral. 45.1xx3dx [Hint: Substitute u = u=x6.]Make a substitution to express the integrand as a rational function and then evaluate the integral. 46.1+xxdxMake a substitution to express the integrand as a rational function and then evaluate the integral. 47.e2xe2x+3ex+2dxMake a substitution to express the integrand as a rational function and then evaluate the integral. 48.sinxcos2x3cosxdxMake a substitution to express the integrand as a rational function and then evaluate the integral. 49.sec2ttan2t+3tant+2dt50EMake a substitution to express the integrand as a rational function and then evaluate the integral. 51.dx1+ex52EUse integration by parts, together with the techniques of this section, to evaluate the integral. 53.ln(x2x+2)dx54EUse a graph of f(x) = 1/(x2 2x 3) to decide whether 02f(x)dx is positive or negative. Use the graph to give a rough estimate of the value of the integral and then use partial fractions to find the exact value.Evaluate 1x2+kdx by considering several cases for the constant k.Evaluate the integral by completing the square and using Formula 6. 57.dxx22x58EThe German mathematician Karl Weierstrass (18151897) noticed that the substitution t = tan(x/2) will convert any rational function of sin x and cos x into an ordinary rational function of t. (a) If t = tan(x/2), x , sketch a right triangle or use trigonometric identities to show that cos(x2)=11+t2andsin(x2)=11+t2 (b) Show that cosx=1x21+t2andsinx=2t1+t2 (b) Show that dx=21+t2dtUse the substitution in Exercise 59 to transform the integrand into a rational function of t and then evaluate the integral. 60.dx1cosx61E62EUse the substitution in Exercise 59 to transform the integrand into a rational function of t and then evaluate the integral. 63.0/2sin2x2+cosxdxFind the area of the region under the given curve from 1 to 2. 64.y=1x3+xFind the area of the region under the given curve from 1 to 2. 65.y=x2+13xx2Find the volume of the resulting solid if the region under the curve. y = 1/(x2 + 3x + 2) from x = 0 to x = 1 is rotated about (a) the x-axis and (b) the y-axis.One method of slowing the growth of an insect population without using pesticides is to introduce into the population a number of sterile males that mate with fertile females but produce no offspring. (The photo shows a screw-worm fly, the first pest effectively eliminated from a region by this method.) Let P represent the number of female insects in a population and S the number of sterile males introduced each generation. Let r be the per capita rate of production of females by females, provided their chosen mate is not sterile. Then the female population is related to time t by t=P+SP[(r1)PS]dP Suppose an insect population with 10,000 females grows at a rate of r = 1.1 and 900 sterile males are added initially. Evaluate the integral to give an equation relating the female population to time. (Note that the resulting equation can't be solved explicitly for P.)68E71E(a) Use integration by parts to show that, for any positive integer n, dx(x2+a2)ndx=x2a2(n1)(x2+a2)n1+2n32a2(n1)dx(x2+a2)n1 (b) Use part (a) to evaluate dx(x2+1)2anddx(x2+1)3Suppose that F, G, and Q are polynomials and F(x)Q(x)=G(x)Q(x) for all x except when Q(x) = 0. Prove that F(x) = G(x) for all x. [Hint: Use continuity.]If f is a quadratic function such that f(0) = 1 and f(x)x2(x+1)3dx is a rational function, find the value of f'(0).If a 0 and n is a positive integer, find the partial fraction decomposition of f(x)=1xn(xa) [Hint: First find the coefficient of 1/(x a). Then subtract the resulting term and simplify what is left.]Evaluate the integral. 1. cosx1sinxdxEvaluate the integral. 2. 01(3x+1)2dxEvaluate the integral. 3. 14ylnydyEvaluate the integral. 4. sin3xcosxdxEvaluate the integral. 5. tt4+2dtEvaluate the integral. 6. 01x(2x+1)3dxEvaluate the integral. 7. 11earctany1+y2dyEvaluate the integral. 8. tsintcostdtEvaluate the integral. 9. 24x+2x2+3x4dxEvaluate the integral. 10. cos(1/x)x3dxEvaluate the integral. 11. 1x3x21dxEvaluate the integral. 12. 2x3x3+3xdxEvaluate the integral. 13. sin5tcos4tdtEvaluate the integral. 14. ln(1+x2)dxEvaluate the integral. 15. xsecxtanxdxEvaluate the integral. 16. 02/2x21x2dxEvaluate the integral. 17. 0tcos2tdt18EEvaluate the integral. 19. ex+exdx20EEvaluate the integral. 21. arctanxdxEvaluate the integral. 22. lnxx1+(lnx)2dxEvaluate the integral. 23. 01(1+x)8dxEvaluate the integral. 24. (1+tanx)2secxdxEvaluate the integral. 25. 011+12t1+3tdtEvaluate the integral. 26. 013x2+1x3+x2+x+1dxEvaluate the integral. 27. dx1+exEvaluate the integral. 28. sinatdtEvaluate the integral. 29. ln(x+x21)dxEvaluate the integral. 30. 12|ex1|dx31E32EEvaluate the integral. 33. 32xx2dxEvaluate the integral. 34. /4/21+4cotx4cotxdx35E36E37E38E39E40E41E42E43EEvaluate the integral. 44. 1+exdxEvaluate the integral. 45. x5ex3dxEvaluate the integral. 46. (x1)exx2dxEvaluate the integral. 47. x3(x1)4dx48EEvaluate the integral. 49. 1x4x+1dx50E51EEvaluate the integral. 52. dxxx4+153E54EEvaluate the integral. 55. dxx+xxEvaluate the integral. 56. dxx+xx57E58E59E60EEvaluate the integral. 61. d1+cos62E63E64E65E66E67E68EEvaluate the integral. 69. 131+x2x2dxEvaluate the integral. 70. 11+2exexdxEvaluate the integral. 71. e2x1+exdx72EEvaluate the integral. 73. x+arcsinx1x2dx74E75E76E77EEvaluate the integral. 78. 1+sinx1sinxdx79E80E81E82E83E84EUse the indicated entry in the Table of Integrals on the Reference Pages to evaluate the integral. 1. 0/2cos5xcos2xdx; entry 80Use the indicated entry in the Table of Integrals on the Reference Pages to evaluate the integral. 2. 01xx2dx; entry 113Use the indicated entry in the Table of Integrals on the Reference Pages to evaluate the integral. 3. 124x23dx; entry 39Use the indicated entry in the Table of Integrals on the Reference Pages to evaluate the integral. 4. 01tan3(x/6)dx; entry 69Use the Table of Integrals on Reference Pages 610 to evaluate the integral. 5. 0/8arctan2xdxUse the Table of Integrals on Reference Pages 610 to evaluate the integral. 6. 02x24x2dxUse the Table of Integrals on Reference Pages 610 to evaluate the integral. 7. cosxsin2x9dxUse the Table of Integrals on Reference Pages 610 to evaluate the integral. 8. ex4e2xdxUse the Table of Integrals on Reference Pages 610 to evaluate the integral. 9. 9x2+4x2dxUse the Table of Integrals on Reference Pages 610 to evaluate the integral. 10. 2y23y2dyUse the Table of Integrals on Reference Pages 610 to evaluate the integral. 11. 0cos6dUse the Table of Integrals on Reference Pages 610 to evaluate the integral. 12. x2+x4dxUse the Table of Integrals on Reference Pages 610 to evaluate the integral. 13. arctanxxdxUse the Table of Integrals on Reference Pages 610 to evaluate the integral. 14. 0x3sinxdxUse the Table of Integrals on Reference Pages 610 to evaluate the integral. 15. coth(1/y)y2dyUse the Table of Integrals on Reference Pages 610 to evaluate the integral. 16. e3te2t1dtUse the Table of Integrals on Reference Pages 610 to evaluate the integral. 17. y6+4y4y2dyUse the Table of Integrals on Reference Pages 610 to evaluate the integral. 18. dx2x33x2Use the Table of Integrals on Reference Pages 610 to evaluate the integral. 19. sin2xcosxln(sinx)dxUse the Table of Integrals on Reference Pages 610 to evaluate the integral. 20. sin25sindUse the Table of Integrals on Reference Pages 610 to evaluate the integral. 21. ex3e2xdxUse the Table of Integrals on Reference Pages 610 to evaluate the integral. 22. 02x34x2x4dxUse the Table of Integrals on Reference Pages 610 to evaluate the integral. 23. sec5xdxUse the Table of Integrals on Reference Pages 610 to evaluate the integral. 24. x3arcsin(x2)dxUse the Table of Integrals on Reference Pages 610 to evaluate the integral. 25. 4+(lnx)2xdxUse the Table of Integrals on Reference Pages 610 to evaluate the integral. 26. 01x4e1dxUse the Table of Integrals on Reference Pages 610 to evaluate the integral. 27. cos1(x2)x3dxUse the Table of Integrals on Reference Pages 610 to evaluate the integral. 28. dx1e2xUse the Table of Integrals on Reference Pages 610 to evaluate the integral. 29. e2x1dxUse the Table of Integrals on Reference Pages 610 to evaluate the integral. 30. etsin(t3)dtUse the Table of Integrals on Reference Pages 610 to evaluate the integral. 31. x4dxx102Use the Table of Integrals on Reference Pages 610 to evaluate the integral. 32. sec2tan29tan2dThe region under the curve y = sin2 x from 0 to is rotated about the x-axis. Find the volume of the resulting solid.Find the volume of the solid obtained when the region under the curve y = arcsin x, x 0, is rotated about the y-axis.Verify Formula 53 in the Table of Integrals (a) by differentiation and (b) by using the substitution t = a + bu.Verify Formula 31 (a) by differentiation and (b) by substituting u = a sin .Let I=04f(x)dx, where f is the function whose graph is shown. (a) Use the graph to find L2, R2, and M2. (b) Are these underestimates or overestimates of I? (c) Use the graph to find T2. How does it compare with I? (d) For any value of n, list the numbers Ln, Rn, Mn, Tn, and I in increasing order.The left, right, Trapezoidal, and Midpoint Rule approximations were used to estimate 02f(x)dx, where f is the function whose graph is shown. The estimates were 0.7811, 0.8675, 0.8632, and 0.9540, and the same number of sub-intervals were used in each case. (a) Which rule produced which estimate? (b) Between which two approximations does the true value of 02f(x)dx lie?Estimate 01cos(x2)dx using (a) the Trapezoidal Rule and (b) the Midpoint Rule, each with n = 4. From a graph of the integrand, decide whether your answers are underestimates or overestimates. What can you conclude about the true value of the integral?Draw the graph of f(x)=sin(12x2) in the viewing rectangle [0, 1] by [0, 0.5] and let I=01f(x)dx. (a) Use the graph to decide whether L2, R2, M2, and T2 underestimate or overestimate I. (b) For any value of n, list the numbers Ln, Rn, Tn, Mn and I in increasing order. (c) Compute L5, R5, M5, and T5. From the graph, which do you think gives the best estimate of I?Use (a) the Midpoint Rule and (b) Simpsons Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.) Compare your results to the actual value to determine the error in each approximation. 1. 02x1+x2dx, n =10Use (a) the Midpoint Rule and (b) Simpsons Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.) Compare your results to the actual value to determine the error in each approximation. 2. 0xcosxdx, n = 4Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpsons Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.) 7. 12x31dx, n = 10Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpsons Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.) 8. 0211+x6dx, n = 8Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpsons Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.) 9. 02ex1+x2dx, n = 10