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Fill in the blanks to complete each of the following theorem statements:

If uand vare functions such that u'(x)v(x)+u(x)v'(x)is _____, then we can use the product rule in reverse to obtainu'(x)v(x)+u(x)v'(x)dx=_____.

Short Answer

Expert verified

If uand vare functions such that u'(x)v(x)+u(x)v'(x)dxis integrable, then we can use the product rule in reverse to obtainu'(x)v(x)+u(x)v'(x)dx=u(x)v(x)+C.

Step by step solution

01

Step 1. Given information 

If uand vare functions such that u'(x)v(x)+u(x)v'(x)dxis _____, then we can use the product rule in reverse to obtain u'(x)v(x)+u(x)v'(x)dx=_____.

02

Step 2. Filling in the blanks to complete the theorem statements 

If uand vare functions such that u'(x)v(x)+u(x)v'(x)dx is integrable, then we can use the product rule in reverse to obtain u'(x)v(x)+u(x)v'(x)dx=u(x)v(x)+C.

By using reverse of product rule, we get the integral above.

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Most popular questions from this chapter

Solve the integral:3xex2dx

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 solving1x24dx.

(b) True or False: The substitution x = 2 sec u is a suitable choice for solving1x24dx.

(c) True or False: The substitution x = 2 tan u is a suitable choice for solving1x2+4dx.

(d) True or False: The substitution x = 2 sin u is a suitable choice for solvingx2+45/2dx

(e) True or False: Trigonometric substitution is a useful strategy for solving any integral that involves an expression of the form x2a2.

(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.

Find three integrals in Exercises 39–74 that can be solved without using trigonometric substitution.

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=tanu.

(b) An integral with which we could reasonably apply trigonometric substitution with x=4secu.

(c) An integral with which we could reasonably apply trigonometric substitution with x2=3sinu.

Solve each of the integrals in Exercises 39–74. Some integrals require trigonometric substitution, and some do not. Write your answers as algebraic functions whenever possible.

13-x2dx

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