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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.

lnx3dx

Short Answer

Expert verified

The result isxlnx3-3xlnx2+6xlnx-6x+C.

Step by step solution

01

Step 1. Given information.

Consider the given information.

lnx3dx

02

Step 2. Find the integral.

Apply the by part's method to solve the integral.

lnx3dx=xlnx3-x3lnx2dx=xlnx3-x·3lnx2·1xdx=xlnx3-3lnx2dx=xlnx3-3xlnx2-2lnxdx=xlnx3-3xlnx2-2xlnx-x+C=xlnx3-3xlnx2-6xlnx+6x+C=xlnx3-3xlnx2+6xlnx-6x+C

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Most popular questions from this chapter

Find three integrals in Exercises 21–70 in which the denominator of the integrand is a good choice for a substitution u(x).

Solvex4+x2dxthe following two ways:

(a) with the substitution u=4+x2;

(b) with the trigonometric substitution x = 2 tan u.

Suppose you use polynomial long division to divide p(x) by q(x), and after doing your calculations you end up with the polynomial x2-x+3 as the quotient above the top line, and the polynomial 3x − 1 at the bottom as the remainder. Thenp(x)=___andp(x)q(x)=____

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 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.

x3x2+1dx

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