In Plancks Radiation Law, we encounter the integral, 1dxx5(e1/x 1).(a) Explain why a

In Exercises 19, use the box on page 404 and the behavior of rational and exponential functions as x to predict whether the integrals converge or diverge. , 1 x2 x4 + 1 dx

In Exercises 19, use the box on page 404 and the behavior of rational and exponential functions as x to predict whether the integrals converge or diverge. , 2 x3 x4 1 dx

In Exercises 19, use the box on page 404 and the behavior of rational and exponential functions as x to predict whether the integrals converge or diverge. , 1 x2 + 1 x3 + 3x + 2 dx

In Exercises 19, use the box on page 404 and the behavior of rational and exponential functions as x to predict whether the integrals converge or diverge. , 1 1 x2 + 5x + 1 dx

In Exercises 19, use the box on page 404 and the behavior of rational and exponential functions as x to predict whether the integrals converge or diverge. , 1 x x2 + 2x + 4 dx

In Exercises 19, use the box on page 404 and the behavior of rational and exponential functions as x to predict whether the integrals converge or diverge. , 1 x2 6x + 1 x2 + 4 dx

In Exercises 19, use the box on page 404 and the behavior of rational and exponential functions as x to predict whether the integrals converge or diverge. , 1 5x + 2 x4 + 8x2 + 4 dx

In Exercises 19, use the box on page 404 and the behavior of rational and exponential functions as x to predict whether the integrals converge or diverge. , 1 1 e5t + 2 dt

In Exercises 19, use the box on page 404 and the behavior of rational and exponential functions as x to predict whether the integrals converge or diverge. , 1 x2 + 4 x4 + 3x2 + 11 dx

In Exercises 1025, decide if the improper integral converges or diverges. , 50 dz z3

In Exercises 1025, decide if the improper integral converges or diverges. , 1 dx 1 + x

In Exercises 1025, decide if the improper integral converges or diverges. , 1 dx x3 + 1

In Exercises 1025, decide if the improper integral converges or diverges. , 8 5 6 t 5 dt

In Exercises 1025, decide if the improper integral converges or diverges. , 1 0 1 x19/20 dx

In Exercises 1025, decide if the improper integral converges or diverges. , 5 1 dt (t + 1)2

In Exercises 1025, decide if the improper integral converges or diverges. , du 1 + u2

In Exercises 1025, decide if the improper integral converges or diverges. , 1 du u + u2

In Exercises 1025, decide if the improper integral converges or diverges. , 1 d 2 + 1

In Exercises 1025, decide if the improper integral converges or diverges. , 2 d 3 + 1

In Exercises 1025, decide if the improper integral converges or diverges. , 1 0 d 3 +

In Exercises 1025, decide if the improper integral converges or diverges. , 0 dy 1 + ey

In Exercises 1025, decide if the improper integral converges or diverges. , 1 2 + cos 2 d

In Exercises 1025, decide if the improper integral converges or diverges. , 0 dz ez + 2z

In Exercises 1025, decide if the improper integral converges or diverges. , 0 2 sin 2 d

In Exercises 1025, decide if the improper integral converges or diverges. , 4 3 + sin d

The graphs of y = 1/x, y = 1/x2 and the functions f(x), g(x), h(x), and k(x) are shown in Figure 7.24. (a) Is the area between y = 1/x and y = 1/x2 on the interval from x = 1 to finite or infinite? Explain. (b) Using the graph, decide whether the integral of each of the functions f(x), g(x), h(x) and k(x) on the interval from x = 1 to converges, diverges, or whether it is impossible to tell.

Suppose - a f(x) dx converges. What does Figure 7.25 suggest about the convergence of - a g(x) dx?

For what values of p do the integrals in Problems 2829 converge or diverge? , 2 dx x(ln x)p

For what values of p do the integrals in Problems 2829 converge or diverge? , 2 1 dx x(ln x)p

(a) Find an upper bound for , 3 e x2 dx. [Hint: ex2 e3x for x 3.] (b) For any positive n, generalize the result of part (a) to find an upper bound for , n ex2 dx by noting that nx x2 for x n

In Plancks Radiation Law, we encounter the integral , 1 dx x5(e1/x 1). (a) Explain why a graph of the tangent line to et at t = 0 tells us that for all t 1 + t e t . (b) Substituting t = 1/x, show that for all x = 0 e 1/x 1 > 1 x. (c) Use the comparison test to show that the original integral converges.

In Problems 3235, explain what is wrong with the statement. - 1 1/(x3 + sin x) dx converges by comparison with - 1 1/x3 dx.

In Problems 3235, explain what is wrong with the statement. - 1 1/(x 2 + 1) dx is divergent.

In Problems 3235, explain what is wrong with the statement. If 0 f(x) g(x) and - 0 g(x) dx diverges then by the comparison test - 0 f(x) dx diverges.

In Problems 3235, explain what is wrong with the statement. Let f(x) > 0. If - 1 f(x) dx is convergent then so is - 1 1/f(x) dx.

In Problems 3637, give an example of: A continuous function f(x) for x 1 such that the improper integral - 1 f(x)dx can be shown to converge by comparison with the integral - 1 3/(2x2) dx.

In Problems 3637, give an example of: A positive, continuous function f(x) such that - 1 f(x)dx diverges and f(x) 3 7x 2 sin x, for x 1.

In Problems 3839, decide whether the statements are true or false. Give an explanation for your answer. The integral , 0 1 ex + x dx converges.

In Problems 3839, decide whether the statements are true or false. Give an explanation for your answer. The integral , 1 0 1 x2 3 dx diverges.