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Chapter 5: Q. 28 (page 417)

Solve each of the integrals in Exercises 21鈥�70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)

鈭�3cos(蟺x)dx

Short Answer

Expert verified

The solution of the given integral is 鈭�3cos(蟺x)dx=3蟺蝉颈苍蟺虫+C.

Step by step solution

01

Step 1. Given Information 

Solving the given integrals.

鈭�3cos(蟺x)dx

02

Step 2. Solving the given integral using substitution method. 

Let

u=蟺xdudx=蟺du=蟺dx1蟺du=dx

03

Step 3. This substitution changes the integral into 

鈭�3cos(蟺x)dx=3蟺鈭�cosudu鈭�3cos(蟺虫)dx=3蟺sinu+C鈭�3cos(蟺虫)dx=3蟺蝉颈苍蟺虫+C

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Most popular questions from this chapter

Solve the integral:鈭�xexdx

Find three integrals in Exercises 21鈥�70 in which the denominator of the integrand is a good choice for a substitution u(x).

Why is it okay to use a triangle without thinking about the unit circle when simplifying expressions that result from a trigonometric substitution withx=asinuor x=atanu? Why do we need to think about the unit circle after trigonometric substitution with x=asecu?

Show that if x=tan鈦�u, then dx=sec2鈦�udu, in the following two ways: (a) by using implicit differentiation, thinking of uas a function of x, and (b) by thinking of xas a function of u.

Solve given definite integral.

鈭�1/41/2鈥�1x21鈭�x2dx
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