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Chapter 5: Q. 20 (page 441)

Calculate each of the integrals in Exercises 17鈥�46. For some integrals you may need to use polynomial long division, partial fractions, factoring or expanding, or the method of completing the square.

鈭�x+1(x-1)3dx

Short Answer

Expert verified

The value is -xx2-2x+1+C

Step by step solution

01

Step 1. Given Information  

The given integral is鈭�x+1(x-1)3dx

02

Step 2. Calculation 

The integral can be rewritten as follows,

鈭�x+1(x-1)3dx=鈭�1x-12+2(x-1)3dx=鈭�1x-12dx+鈭�2(x-1)3dx

role="math" localid="1648812682438" Letu=x-1,du=dx鈭�1(x-1)2dx+鈭�2(x-1)3dx=鈭�1u2du+鈭�2u3du=u-1-1+2u-2-2+C=-1u-1u2+C=-1x-1-1x-12+C=-xx2-2x+1+C

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Most popular questions from this chapter

Consider the integral 鈭�1x21鈭�x2dxfrom the reading at the beginning of the section.

(a) Use the inverse trigonometric substitution u=sin鈭�1鈦�xto solve this integral.

(b) Use the trigonometric substitution x=sin鈦�u to solve the integral.

(c) Compare and contrast the two methods used in parts (a) and (b).

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

鈭�9-x2xdx

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.

Solve the integral:鈭�3xex2dx

Solve the integral:鈭�xexdx
See all solutions

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