Integrating 1/(1+tan(x)^e) from 0 to pi/2: Challenges and Possible Approaches

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In summary: for all n>0 and this is true for all e except for 1 and e = 10. so for all those e, the integral has the same value as the numerically determined e.
  • #1
iceblits
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Integrate(1/(1 + [ tan x ] ^e)) from 0 to pi/2

I don't think there's an anti derivative so some other method has to be used to get an exact answer (no approximation). I've tried using Taylor series and Eulers formula. Any help would be great..maybe if you could just point me in the right direction even...
 
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  • #2
Is that denominator ...
1 + [ tan x^e ]
or
1 + [ tan x ] ^e ?
 
  • #3
its 1 + [ tan x ] ^e
 
  • #4
Ok I think I got it..I think its pi/4.
 
  • #5
Split the integral up in two parts, from 0 to pi/4 and pi/4 to pi/2. Then use the substitution u = pi/2-x on the second part. Then you will see that it is much easier to compute. How did you arrive upon pi/4? I'm curious.
 
  • #6
i just used rectangle approximation and it seemed that it would equal pi/4 (i knew that it had to be an exact answer because i was told so)..your way is probably going to work nicely as well..i will try it..
 
  • #7
Ok I tried that and I don't really see where its going. The integral doesn't seem to be computable by finding an anti derivative
 
  • #8
iceblits said:
Integrate(1/(1 + [ tan x ] ^e)) from 0 to pi/2

I don't think there's an anti derivative so some other method has to be used to get an exact answer (no approximation). I've tried using Taylor series and Eulers formula. Any help would be great..maybe if you could just point me in the right direction even...

What is e? Is it the standard base of natural logarithms, or just some other (positive?) number?
Using Maple I have found that for e = integer 1,...,10, the integral has the same value for all such e, and (by numerical integration), has the same value also for non-integer e. However, at present I don't understand why this happens. You could use Wolfram Alpha to verify this for yourself.

RGV
 
  • #9
woah..interesting...I will check that out and yes its e as in the 2.71828183...
 
  • #10
wowwwww..thats so cool.. i noticed that before but i didnt think it was the exact same value (i thought it might be off by a few decimal points or something because the region was so small, but using mathematica i can confirm what u said)
 
  • #11
I think its because tangent is a relationship between the sin and the cos so the ratio levels out
 
  • #12
edit: nevermind i don't know why it is yet..but I intend to find out :)
 
  • #13
ok the reason y it does that is because 1/(1+tan(x)^n) is has integral zero over the line y=-pi/2(x)+1
 

1. How do you find the limits of integration?

To find the limits of integration, first determine the independent variable in the integral. Then, look at the given function and determine the values of the independent variable where the function is defined. These values will be the limits of integration.

2. How do you choose the appropriate method to solve the integral?

The appropriate method to solve an integral depends on the form of the function and the properties of the integral. Some commonly used methods include substitution, integration by parts, and trigonometric identities. It is important to familiarize yourself with these methods and practice using them to determine the most efficient and effective approach for a given integral.

3. How do you handle indefinite integrals?

Indefinite integrals do not have upper and lower limits like definite integrals. To solve an indefinite integral, you must find an expression that, when differentiated, gives the original function. This expression is known as the antiderivative. Remember to always add a constant of integration when solving indefinite integrals.

4. What are some common integration techniques?

Some common integration techniques include integration by substitution, integration by parts, partial fractions, and trigonometric identities. Other techniques such as u-substitution and integration by trigonometric substitution may also be useful for specific types of integrals. It is important to practice using these techniques to become comfortable with them.

5. How do you check your answer for an integral?

To check your answer for an integral, you can differentiate the antiderivative you found to see if it gives the original function. You can also use a graphing calculator or software to graph both the original function and the antiderivative and visually compare them. Additionally, you can use definite integrals to check your answer by plugging in the limits of integration and comparing the results to the original function's integral.

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