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

x+1(x-1)2dx

Short Answer

Expert verified

The value is1x-1((x-1)ln|x-1|-2)+C

Step by step solution

01

Step 1. Given Information  

The given integral is x+1(x-1)2dx

02

Step 2. Calculation 

The integral can be rewritten as follows,

x+1(x-1)2dx=1x-1+2(x-1)2dx=1x-1dx+2(x-1)2dx

Letu=x-1,du=dx1(x-1)dx+2(x-1)2dx=1udu+2u2du=ln|u|+2u-1-1+C=ln|x-1|-2x-1+C=1x-1x-1ln|x-1|-2+C

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Most popular questions from this chapter

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

2(x+2)2

Explain why 2xx2+1dxand 1xlnxdxare essentially the same integral after a change of variables.

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

x2+6x2

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 given definite integrals.

04xx2+4dx

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