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Chapter 5: Q 3 (page 453)

Trigonometric substitution: Some integrals that don鈥檛 start out involving trigonometric functions can be cleverly converted into ones that do, and then we can bring the techniques of this section to bear. In the next few problems this is how we will solve the integral

鈭�dxx1-x2

Short Answer

Expert verified

It is proved that鈭�dxx1-x2=鈭�cscudu

Step by step solution

01

Step 1. Given information

An integral is鈭�dxx1-x2

02

Step 2. Explanation

鈭�dxx1-x2=鈭�cosudusinu1-sin2u=鈭�cosudusinucosu=鈭�dusinu=鈭�cscdu

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Most popular questions from this chapter

Suppose you use polynomial long division to divide p(x) by q(x), and after doing your calculations you end up with the polynomial x2-x+3 as the quotient above the top line, and the polynomial 3x 鈭� 1 at the bottom as the remainder. Thenp(x)=___andp(x)q(x)=____

For each integral in Exercises 5鈥�8, write down three integrals that will have that form after a substitution of variables.

鈭�sinudu

Solve鈭�x21+x2dx the following two ways:

(a) with the substitution u=tan-1x;

(b) with the trigonometric substitution x = tan u.

Solve the integral: 鈭�xsinxdx

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鈭�1x2鈭�4dx.

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

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

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

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

(f) True or False: Trigonometric substitution doesn鈥檛 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.
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