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Calculus/Improper Integrals

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Expert: - 6/1/2008


Question
Hi Paul,
I have not done an integral with infinite bounds so far. Could you tell me how to go about evaluating this type of an integral.
   Evaluate integral -inf to inf (1/e^x + e^-x )dx
Thank you so much!!

Answer

arctan(x)
Questioner:   Sami
Category:  Calculus
Private:  No
 
Subject:  integral with infinite bounds
Question:  Hi Paul,
I have not done an integral with infinite bounds

>> called an Improper Integral.

so far. Could you tell me how to go about evaluating this type of an integral.
Evaluate integral -inf to inf (1/e^x + e^-x )dx
Thank you so much!!
..................................
Hi, Sami,

In general, you start by writing:

{b
|  (your integrand) dx
}a

Then write:

(value at b) - (value at a)

Let b -> +inf for the first term,
Let a -> -inf for the second term.

OR YOU CAN DO:
{b
|  (your integrand) dx
}-b

and just let b -> inf for both terms.
...........................

In this case, you have

{b      1
|  ------------- dx
}-b  e^x + e^-x

= arctan(e^x) from -b to b

Don't know how to do that?  Try multiplying top and bottom by  e^x.  You get:

{b       e^x
|  ------------- dx
}a  (e^x)^2 + 1

Now do a  u = e^x and get:

{b      1
|  --------- du = arctan(u)
}a  u^2 + 1

[I cheated. I used THE INTEGRATOR. Much faster.  Even I get lazy now and then.]
...................................
Back to work:

= arctan(e^x) from -b to b

= arctan(e^b) - arctan(e^-b)

Now if b -> inf,  e^b --> inf. and e^-b -> 0

But arctan(inf) = pi/2  [look at the graph.]
and arctan(0) = 0

So you have  pi/2 - 0 = pi/2

    Questioner's Rating
    Rating(1-10)Knowledgeability = 10Clarity of Response = 10Politeness = 10
    CommentWow Paul, you are a genius!!! Also that little graph at the top of the response was a helpful reference. Thanks so much for helping


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Paul Klarreich

Expertise

All topics in first-year calculus including infinite series, max-min and related rate problems. Also trigonometry and complex numbers, theory of equations, exponential and logarithmic functions. I can also try (but not guarantee) to answer questions on Analysis -- sequences, limits, continuity.

Experience

I taught all mathematics subjects from elementary algebra to differential equations at a two-year college in New York City for 25 years.

Education/Credentials
(See above.)

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