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Chapter 5: Q. 47 (page 442)

Calculate each of the definite integrals in Exercises 47鈥�52. Some integrals require partial fractions or polynomial long division, and some do not.

鈭�344x2-4dx

Short Answer

Expert verified

The value is-ln(3)+ln(5)

Step by step solution

01

Step 1. Given Information:

Given definite integral 鈭�344x2-4dx

We want to calculate each of the definite integrals.

02

Step 2. Calculation:

We have

鈭�344x2-4dx=4鈭�341x2-4dxSubstitutionx=2uWhenx=4thenu=2andwhenx=3thenu=32andalsodx=2duApplyIntegralSubstitution:4鈭�32214(u2-1)2du=2鈭�3221(u2-1)du=2鈭�3221-(-u2+1)du=-2鈭�32211-u2du=-2lnu+12-lnu-12232=-lnu+1-lnu-1232=-ln|3|-ln|1|-ln|52|-ln|12|Uselogproperties:=ln(3)-ln(52)-ln(2)=-ln(3)+ln(5)

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Most popular questions from this chapter

Complete the square for each quadratic in Exercises 28鈥�33. Then describe the trigonometric substitution that would be appropriate if you were solving an integral that involved that quadratic.

x2鈭�4x鈭�8

Describe two ways in which the long-division algorithm for polynomials is similar to the long-division algorithm for integers and then two ways in which the two algorithms are different.

What is a rational function? What does it mean for a rational function to be proper? Improper?

Solve the integral

鈭�4-xx-2dx

For each function u(x) in Exercises 9鈥�12, write the differential du in terms of the differential dx.

u(x)=x2+1
See all solutions

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