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Chapter 5: Q. 49 (page 478)

Use limits of definite integrals to calculate each of the improper integrals in Exercises 21鈥�56.

鈭�e鈭�1x(lnx)2dx

Short Answer

Expert verified

The value is1(Convergent).

Step by step solution

01

Step 1. Given Information.

The given integral is鈭�e鈭�1x(lnx)2dx.

02

Step 2. Value of the limit.

On simplifying,

鈭�e鈭�1x(lnx)2dx

role="math" localid="1648888264087" Let,u=logxOndifferentiating,dx=xduNow,=鈭�u=1u=鈭�1xu2xdu=1-1鈭�1鈭�u-2du=-1u鈭�1=-1鈭�--11=1(converges)

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Most popular questions from this chapter

Show that if x=tan鈦�u, then dx=sec2鈦�udu, in the following two ways: (a) by using implicit differentiation, thinking of uas a function of x, and (b) by thinking of xas a function of u.

Solve 鈭�x+2x2+4x32dxthe following two ways:

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

(b) by completing the square and then applying the trigonometric substitution x + 2 = 2 sec u.

Solve each of the integrals in Exercises 39鈥�74. Some integrals require trigonometric substitution, and some do not. Write your answers as algebraic functions whenever possible.

鈭�1x2+432dx

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

List some things which would suggest that a certain substitution u(x) could be a useful choice. What do you look for when choosing u(x)?
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