Answer: Exploration Consider the integral where is a positive integer. (a) Is the

Chapter 8, Problem 92

(choose chapter or problem)

Exploration Consider the integral

\(\int_{0}^{\pi / 2} \frac{4}{1+(\tan x)^{n}} d x\)

where n is a positive integer.

(a) Is the integral improper? Explain.

(b) Use a graphing utility to graph the integrand for n = 2,4, 8, and 12.

(c) Use the graphs to approximate the integral as \(n \rightarrow \infty\).

(d) Use a computer algebra system to evaluate the integral for the values of n in part (b). Make a conjecture about the value of the integral for any positive integer n. Compare your results with your answer in part (c).

Text Transcription:

Int_0^pi/2 4/1 + (tan x)^n dx

n rightarrow infinity