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Solve each of the integrals in Exercises 21–70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)

231xlnxdx

Short Answer

Expert verified

The solution of the given integral is 231xlnxdx=2ln3-ln2.

Step by step solution

01

Step 1. Given Information 

Solving the given integrals.

231xlnxdx

02

Step 2. Using the substitution method. 

Let

u=lnxdudx=1xdu=1xdx

03

Step 3. Using the information in equations, we can change variables completely:

231xlnxdx=x=2x=31udu231xlnxdx=x=2x=31u1/2du231xlnxdx=x=2x=3u-1/2du231xlnxdx=u-1/2+1-1/2+1x=2x=3231xlnxdx=u1/21/2x=2x=3231xlnxdx=2u23231xlnxdx=2lnxx=2x=3231xlnxdx=2ln3-ln2

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Most popular questions from this chapter

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.

1x2+432dx

Find three integrals in Exercises 21–70 that we can anti-differentiate immediately after algebraic simplification.

True/False: Determinewhethereachofthestatementsthat follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: f(x)=x+1x-1is a proper rational function.

(b) True or False: Every improper rational function can be expressed as the sum of a polynomial and a proper rational function.

(c) True or False: After polynomial long division of p(x) by q(x), the remainder r(x) has a degree strictly less than the degree of q(x).

(d) True or False: Polynomial long division can be used to divide two polynomials of the same degree.

(e) True or False: If a rational function is improper, then polynomial long division must be applied before using the method of partial fractions.

(f) True or False: The partial-fraction decomposition of x2+1x2(x-3)is of the form Ax2+Bx-3

(g) True or False: The partial-fraction decomposition of x2+1x2(x-3)is of the form Bx+Cx2+Ax-3.

(h) True or False: Every quadratic function can be written in the formA(x-k)2+C

Solve the integral:x2exdx

Consider the integral 1x21x2dxfrom the reading at the beginning of the section.

(a) Use the inverse trigonometric substitution u=sin1xto solve this integral.

(b) Use the trigonometric substitution x=sinu to solve the integral.

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

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