Improper Integrals with u sub and integration by parts ?

In summary, the problem asks to find the integral from 1 to infinity of (lnx)/(x) dx using u substitution and integration by parts. After some discussion, it is determined that u substitution is enough to solve the integral, with u = ln(x) and du = 1/x dx. To determine convergence, the integral is evaluated between x=0 and x=M and taking the limit as M approaches infinity.
  • #1
Jay J
26
0

Homework Statement



The integral from 1 to infinity of (lnx)/(x) dx

Homework Equations


U substitution and integration by parts


The Attempt at a Solution


Cant decide what to use as my "u" . . can anyone help with this part ?

-Jay J-
 
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  • #2
this doesn't really require integration by parts, just a u-substitution:

[tex]\int \frac{ln(x)}{x}=\int u du = \frac{1}{2}u+ C= \frac{1}{2}(ln(x))^2+C[/tex]

here you take u = ln(x) and differentiate to get, du = 1/x dx.
 
  • #3
ok i understand the whole u substitution part but then i have to determine whether or not the integral is divergent or convergent and if its convergent i have 2 evaluate it . .. how would you do that ?

-Jay J-
 
  • #4
Evaluate the integral between x=0 and x=M and let M approach infinity. What do you conclude?
 

1. What is the u-substitution method for improper integrals?

The u-substitution method, also known as the change of variables method, is a technique used to evaluate improper integrals by substituting a new variable for the original variable in the integrand. This often simplifies the integral and makes it easier to evaluate.

2. When should I use u-substitution for improper integrals?

U-substitution is most useful when the integrand contains a complicated expression or a function raised to a power. It is also helpful when the integrand contains a product of functions or when the limits of integration are infinite.

3. Can u-substitution be used for all types of improper integrals?

No, u-substitution can only be used for certain types of improper integrals. It is most commonly used for integrals with infinite limits of integration or integrals with a discontinuous integrand.

4. What is the integration by parts method for improper integrals?

The integration by parts method is another technique used to evaluate improper integrals. It involves breaking down the integrand into two parts and using the product rule of differentiation to rewrite the integral in a different form that is easier to evaluate.

5. When should I use integration by parts for improper integrals?

Integration by parts is most useful when the integrand contains a product of functions, especially if one of the functions is a polynomial or a trigonometric function. It is also helpful when the integral involves logarithmic or exponential functions.

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