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Chapter 5: Q. 50 (page 478)

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

鈭�1e1x(lnx)2dx

Short Answer

Expert verified

The answer is鈭�(Diverges).

Step by step solution

01

Step 1. Given information.

The given integral is鈭�1e1x(lnx)2dx.

02

Step 2. Substitution.

Now,

Letu=ln(x)i.edu=1xdx鈭�1x(ln(x))2dx=鈭�1u2du=鈭�u-2du=-1u鈭�1x(ln(x))2dx=-1ln(x)

03

Step 3. Value of the integral.

Now, by definition,

鈭�1e1x(lnx)2dx=limA鈫�1鈭�Ae1x(lnx)2dx=limA鈫�1-1ln(x)Ae=-limA鈫�11ln(e)+limA鈫�11ln(A)=limA鈫�111+1ln(1)=1+鈭�=鈭�Diverges.

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