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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.


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

Short Answer

Expert verified

The value islnx-2-lnx+1+C

Step by step solution

01

Step 1. Given Information

The given integral is3(x-2)(x+1)dx

02

Step 2. Calculation

The integral can be rewritten as 31(x-2)(x+1)dx

The partial fraction of decomposition is as follows,

1(x-2)(x+1)=Ax+1+Bx-2=A(x-2)+B(x+1)x+1x-2

So,

1=A(x-2)+B(x+1)1=Ax-2A+Bx+B1=x(A+B)-2A+B

The following system of equations are,

A+B=0-2A+B=1Onsolving,weget,A=-13,B=13

03

Step 3. Calculation

The partial fraction can be written as follows,

1(x-2)(x+1)=Ax+1+Bx-2=-13x+1+13x-2=13x-6-13x+3

Now, the integration is solved below,

313x-6+13x-3dx=1x-2dx-1x+1dx=ln|x-2|-ln|x+1|+C

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Most popular questions from this chapter

Why is it okay to use a triangle without thinking about the unit circle when simplifying expressions that result from a trigonometric substitution withx=asinuor x=atanu? Why do we need to think about the unit circle after trigonometric substitution with x=asecu?

Solvex21+x2dx the following two ways:

(a) with the substitution u=tan-1x;

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

Why doesn’t the definite integral231-x2dx make sense? (Hint: Think about domains.)

Solve the integral:x2e3xdx.

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 solving1x24dx.

(b) True or False: The substitution x = 2 sec u is a suitable choice for solving1x24dx.

(c) True or False: The substitution x = 2 tan u is a suitable choice for solving1x2+4dx.

(d) True or False: The substitution x = 2 sin u is a suitable choice for solvingx2+45/2dx

(e) True or False: Trigonometric substitution is a useful strategy for solving any integral that involves an expression of the form x2a2.

(f) True or False: Trigonometric substitution doesn’t 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.

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