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Chapter 5: Q. 65 (page 452)

Solve each of the integrals in Exercises 21-66. Some of the integrals require the methods presented in this section, and some do not. (The last four exercises involve hyperbolic functions).

鈭�tanhxcosh7xdx.

Short Answer

Expert verified

The value of鈭�tanhxcosh7xdxis,cosh7x7+C.

Step by step solution

01

Step 1. Given information

鈭�tanhxcosh7xdx.

02

Step 2. Let u=cosh x,du=sinh x.dx.

The integration is solved below.

鈭�tanhxcosh7xdx=鈭�sinhxcoshxcosh7xdx=鈭�sinhxcosh6xdx=鈭�u6.du=u77+C=coshx77+C=cosh7x7+C

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Most popular questions from this chapter

Complete the square for each quadratic in Exercises 28鈥�33. Then describe the trigonometric substitution that would be appropriate if you were solving an integral that involved that quadratic.

x鈭�3x2

What is a rational function? What does it mean for a rational function to be proper? Improper?

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) An integral with which we could reasonably apply trigonometric substitution with x=tan鈦�u.

(b) An integral with which we could reasonably apply trigonometric substitution with x=4sec鈦�u.

(c) An integral with which we could reasonably apply trigonometric substitution with x鈭�2=3sin鈦�u.

Solve the integral:鈭�3-xe2xdx

For each function u(x) in Exercises 9鈥�12, write the differential du in terms of the differential dx.

u(x)=x2+1
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