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mathematics
calculus early transcendentals 9th
Calculus Early Transcendentals 9th Edition James Stewart, Daniel K. Clegg, Saleem Watson, Lothar Redlin - Solutions
Make a substitution to express the integrand as a rational function and then evaluate the integral. /1 + x X - dx
Evaluate the integral. (/4 1 - cos 40 de
Evaluate the integral. ax .3 1 + x
Evaluate the integral. Si t sint dt
Evaluate the integral. SVī+ e* dx
Evaluate the integral or show that it is divergent. In x dx
Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the function and its antiderivative (take C − 0).∫ x2 sin 2x dx
Make a substitution to express the integrand as a rational function and then evaluate the integral. sin x dx cos?x – 3 cos x
Evaluate the integral.∫ x sec x tan x dx
Evaluate the integral. dx 1+ Vx
(a) Use the reduction formula to show that(b) Use part (a) and the reduction formula to evaluate ∫ sin4x dx. S sin 2x + C 4 sin?x dx
Make a substitution to express the integrand as a rational function and then evaluate the integral. sec?t dt tan?t + 3 tan t + 2
Evaluate the integral.∫ x tan2x dx
Make a substitution to express the integrand as a rational function and then evaluate the integral. et dx J (e - 2)(e2 + 1)
Evaluate the integral.∫ x sin3x dx
Evaluate the integral. (x – 1)e* dx x?
(a) If g(x) = (sin2x)/x2, use a calculator or computer to make a table of approximate values of ∫t1 g(x) dx for t = 2, 5, 10, 100, 1000, and 10,000. Does it appear that ∫∞1 g(x) dx is convergent?(b) Use the Comparison Theorem with f(x) = 1/x2 to show that ∫∞1 g(x) dx is convergent.(c)
Make a substitution to express the integrand as a rational function and then evaluate the integral. dx 1 + e*
Evaluate the integral. dx cos x 1 -
Evaluate the integral.∫ x3(x – 1)–4 dx
Prove that, for even powers of sine, T/2 sin2"x dx 1:3. 5. ..*. (2n – 1) T 2.4. 6. .. · 2n 2
Evaluate the integral. 1 sec e + 1
Evaluate the integral. xv2 - I x
Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the integrand and its antiderivative (taking C = 0).∫ x sin2(x2) dx
Evaluate the integral. 1 x/4x + 1
Use the Comparison Theorem to determine whether the integral is convergent or divergent. 1+ sin?x
Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the integrand and its antiderivative (taking C = 0). I sinx cosx dx
Evaluate the integral. 1 dx x'/4x + 1
Use the Comparison Theorem to determine whether the integral is convergent or divergent. 1 dx J2 x - In x
Evaluate the integral. 1 dx x/4x² + 1
Use the Comparison Theorem to determine whether the integral is convergent or divergent. p- arctan x Jo 2 + e*
Evaluate the integral or show that it is divergent. tan -x dx .2 8
Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the integrand and its antiderivative (taking C = 0). S sec (x) dx
Evaluate the integral. dx x(x* + 1)
Use Exercise 57 to find ∫ (ln x)3 dx.Data From Exercise 57:Use integration by parts to prove the reduction formula. S In x)" dx = x(In x)" – n (In x)- dx %3D
Evaluate the integral.∫ x2 sinh mx dx
Use the Comparison Theorem to determine whether the integral is convergent or divergent. 2 + cos x Vx* + x?
Use Exercise 58 to find ∫ x4ex dx.Data From Exercise 58:Use integration by parts to prove the reduction formula. 'e*dx %3|
Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the function and its antiderivative (take C = 0). .3 Vx? + 1
Evaluate the integral.∫ (x + sin x)2 dx
Use the Comparison Theorem to determine whether the integral is convergent or divergent. sec?x =dx
Evaluate the integral. dx x + xVx
Find the area of the region bounded by the given curves.y = x2e–x, y = xe–x
Evaluate the integral. dx Vx + xx
The integral ∫∞a f(x) dx is improper because the interval [a, ∞) is infinite. If f has an infinite discontinuity at a, then the integral is improper for a second reason. In this case we evaluate the integral by expressing it as a sum of improper integrals of Type 2 and Type 1 as
Use a graph to find approximate x-coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves. y = arcsin(x), y= 2 - x?
Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral. |V4x? – 4x – 3 dx
Find the area of the region bounded by the given curves.y = sin2 x, y = sin3 x, 0 ≤ x ≤ π
Evaluate the integral. xVx + c dx
The integral ∫∞a f(x) dx is improper because the interval [a, ∞) is infinite. If f has an infinite discontinuity at a, then the integral is improper for a second reason. In this case we evaluate the integral by expressing it as a sum of improper integrals of Type 2 and Type 1 as
Use a graph to find approximate x-coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves.y = x ln(x + 1), y = 3x – x2
Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral.∫ csc5t dt
Find the area of the region bounded by the given curves.y = tan x, y = tan2 x, 0 ≤ x ≤ π/4
Evaluate the integral. x In x dx Vx? – 1
The integral ∫∞a f(x) dx is improper because the interval [a, ∞) is infinite. If f has an infinite discontinuity at a, then the integral is improper for a second reason. In this case we evaluate the integral by expressing it as a sum of improper integrals of Type 2 and Type 1 as
Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the curves about the given axis.y = cos(πx/2), y = 0, 0 ≤ x ≤ 1; about the y-axis
Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral. cos x V4 + sin?x dx
Evaluate the integral. dx x* - 16
The integral ∫∞a f(x) dx is improper because the interval [a, ∞) is infinite. If f has an infinite discontinuity at a, then the integral is improper for a second reason. In this case we evaluate the integral by expressing it as a sum of improper integrals of Type 2 and Type 1 as
Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the curves about the given axis.y = ex, y = e–x, x = 1; about the y-axis
Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral. cot x =dx V1 + 2 sin x
Evaluate the integral. dx x'V4x? – 1 .2
Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the curves about the given axis.y = e–x, y = 0, x = –1, x = 0; about x = 1
Evaluate the integral. de 1 + cos 0
Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the curves about the given axis.y = ex, x = 0, y = 3; about the x-axis
Evaluate the integral. de 1 + cos?e
Use(a) The Trapezoidal Rule(b) The Midpoint Rule(c) Simpson’s Rule with n = 10 to approximate the given integral. Round your answers to six decimal places. 1 ах 12 In x
Is it possible to find a number n such that ∫∞0 xn dx is convergent?
Evaluate the integral. eva dx
(a) Evaluate the integral(b) Guess the value ofwhen n is an arbitrary positive integer.(c) Prove your guess using mathematical induction. S x"e*dx for n = 0, 1, 2, and 3.
Find the volume obtained by rotating the region bounded by the curves about the given axis.y = sec x, y = cos x, 0 ≤ x ≤ π/3; about y = −1
Evaluate the integral. 1 dx Vx + 1
The Cauchy principal value of the integraldefined byShow thatdiverges but the Cauchy principal value of this integral is 0. f(x) dx -00
Evaluate the integral. sin 2x 1+ cos*x
Use(a) The Trapezoidal Rule(b) The Midpoint Rule(c) Simpson’s Rule with n = 10 to approximate the given integral. Round your answers to six decimal places. SVi cos. cos x dx
(a) Use a computer algebra system to find the partial fraction decomposition of the function(b) Use part (a) to find ∫ f(x) dx and graph f and its indefinite integral on the same screen.(c) Use the graph of f to discover the main features of the graph of ∫ f (x) dx. 12x – 7x - 13x2 + 8 100x -
Evaluate the integral. =/3 In(tan x) -dx T/4 sin x cos x
Evaluate the integral. 1 dx Vx +1 + Vx
If f(0) = g(0) = 0 and f" and g" are continuous, show that ()g"(x) dx = f(a)g'(a) – f'(a)g(a) + [f"(x)g(x) dx
(a) Use integration by parts to show that, for any positive integer n,(b) Use part (a) to evaluate dx (x² + a²)" 2a (n – 1)(x + a?)"- 2n – 3 dx 2a (n – 1) J (x? + a?)"-1
Evaluate the integral. x? x° + 3x3 + 2
Evaluate the integral. V1 + x? dx x? Len
Evaluate the integral. 1 1 + 2e* - e*
(a) Recall that the formula for integration by parts is obtained from the Product Rule. Use similar reasoning to obtain the following integration formula from the Quotient Rule.(b) Use the formula in part (a) to evaluate и dv 2 + du
Evaluate the integral. e2x ax 1+ e*
As we saw a radioactive substance decays exponentially: The mass at time t is m(t) = m(0)ekt, where ms0d is the initial mass and k is a negative constant. The mean life M of an atom in the substance isFor the radioactive carbon isotope, 14C, used in radiocarbon dating, the value of k is
The Wallis Product Formula for π Let(a) Show that I2n+2 ≤ I2n+1 ≤ I2n.(b) Use Exercise 56 to show that(c) Use parts (a) and (b) to show thatand deduce that(d) Use part (c) and Exercises 55 and 56 to show thatThis formula is usually written as an infinite product:and is called the Wallis
Evaluate the integral. In(x + 1) dx x?
Evaluate the integral. x + arcsin x V1 - x2
Evaluate the integral. 4* + 10* dx 2*
Evaluate the integral. dx x In x - x
Evaluate the integral. x? dx x² +1
Evaluate the integral. xe* -dx V1 + e*
Evaluate the integral. 1 + sin x 1- sin x
Evaluate the integral.∫ x sin2x cos x dx
Evaluate the integral. sec x cos 2x -dx sin x + sec x
Evaluate the integral. sin x cos x J sin*x + cos*x
Evaluate the integral. SVi - sin x dx
Evaluate the integral. 9- x V dx 9 - x
Evaluate the integral. 1 dx (sin x + cos x)
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