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Chapter 5: Q 26. (page 429)

Consider the integral 鈭�x(x+1)100dx.

(a) Solve this integral by using integration by parts with u = x and dv=(x+1)100dx.

(b) Now solve the integral another way, by using u-substitution with u = x + 1 and back-substitution.

(c) How must your answers to parts (a) and (b) be related? Use graphs to prove this relationship.

Short Answer

Expert verified

(a)lnxlnx-1

(b) lnx

(c) Answers in parts (a), (b) are equal.

Step by step solution

01

Part (a) Step 1. Given information

Integral is鈭�xx+1100dx

02

Part (a) Step 2. Explanation

Takeu=x,dv=x+1100dx

鈭�xx+1100dx=xx+1101101-鈭�x+1101101dx=xx+1101101-x+110210302+C
03

Part (b) Step 1. Explanation

Letu=x+1

role="math" localid="1651754375754" 鈭�xx+1100dx=鈭�u-1u100du=鈭�u101-u100du=u102102-u101101+C=xx+1101101-x+110210302+C
04

Part (c) Step 1. Explanation

Answers in parts (a), (b) are equal.

So, answers are same.

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Most popular questions from this chapter

Solve鈭�x4+x2dxthe following two ways:

(a) with the substitution u=4+x2;

(b) with the trigonometric substitution x = 2 tan u.

Solve the integral:鈭�3x+1secxdx

Show by differentiating (and then using algebra) that 鈭�cot鈦�sin鈭�1鈦�xand 鈭�1鈭�x2xare both antiderivatives of 1x21鈭�x2. How can these two very different-looking functions be an antiderivative of the same function?

Why don鈥檛 we need to have a square root involved in order to apply trigonometric substitution with the tangent? In other words, why can we use the substitution x=atan鈦�uwhen we see x2+a2, even though we can鈥檛 use the substitution x=asin鈦�uunless the integrand involves the square root ofa2鈭�x2? (Hint: Think about domains.)

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.
See all solutions

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