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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 ∫dvand vis that differentiation and integration are inverse operations of each other.

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Most popular questions from this chapter

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:∫xcosxdx.

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The substitution x = 2 sec u is a suitable choice for solving∫1x2−4dx.

(b) True or False: The substitution x = 2 sec u is a suitable choice for solving∫1x2−4dx.

(c) True or False: The substitution x = 2 tan u is a suitable choice for solving∫1x2+4dx.

(d) True or False: The substitution x = 2 sin u is a suitable choice for solving∫x2+4−5/2dx

(e) True or False: Trigonometric substitution is a useful strategy for solving any integral that involves an expression of the form x2−a2.

(f) True or False: Trigonometric substitution doesn’t solve an integral; rather, it helps you rewrite integrals as ones that are easier to solve by other methods.

(g) True or False: When using trigonometric substitution with x=asinu, we must consider the cases x>a and x<-a separately.

(h) True or False: When using trigonometric substitution with x=asecu, we must consider the cases x>a and x<-a separately.

Solve ∫1x2+9dx the following two ways:

(a) with the trigonometric substitution x = 3 tan u;

(b) with algebra and the derivative of the arctangent.

Consider the integral ∫x(x2−1)2dx.

(a) Solve this integral by using u-substitution.

(b) Solve the integral another way, using algebra to multiply out the integrand first.

(c) How must your two answers be related? Use algebra to prove this relationship.
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