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Use whatever method you like to solve each of the given definite and indefinite integrals. These integrals are neither in order of difficulty nor in order of technique. Many of the integrals can be solved in more than one way.

ln2xx2dx

Short Answer

Expert verified

The result is-ln2xx-1x+C.

Step by step solution

01

Step 1. Given information.

Consider the given information.

ln2xx2dx

02

Step 2. Solve the integration.

Apply the integration by parts method to find the solution.

ln2xx2dx=-ln2xx--2x2xdx=-ln2xx+1x2dx=-ln2xx-1x+C

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Most popular questions from this chapter

Write sin(2cos-1x)as an algebraic function.

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

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

u(x)=x2+1

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.

Solve given integrals by using polynomial long division to rewrite the integrand. This is one way that you can sometimes avoid using trigonometric substitution; moreover, sometimes it works when trigonometric substitution does not apply.

x3-3x2+2x-3x2+1dx

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