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Chapter 5: Q. 8 (page 428)

Suppose v(x) is a function of x. Explain why the integral

of dv is equal to v (up to a constant).

Short Answer

Expert verified

Differentiation and integration are inverse operations of each other.

Step by step solution

01

Step 1. Given information

It is given that vxis a function of x.

02

Step 2. Explanation

The derivative of a function vxwill be dvand integration of dvwith respect to x gives back the function vxplus a constant because the integration of a derivative with respect to the same variable results in the function itself. So, the reason of equality of 鈭�dvand vis that differentiation and integration are inverse operations of each other.

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Most popular questions from this chapter

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

u(x)=sinx

Explain how to know when to use the trigonometric substitutions x=asin鈦�u,x=atan鈦�u,andx=asec鈦�u, Describe the trigonometric identity and the triangle that will be needed in each case. What are the possible values for xand uin each case?

List some things which would suggest that a certain substitution u(x) could be a useful choice. What do you look for when choosing u(x)?

Solve the integral:鈭�xexdx

Give an example of an integral for which trigonometric substitution is possible but an easier method is available. Then give an example of an integral that we still don鈥檛 know how to solve given the techniques we know at this point.
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