Integration Problem (math genius/expert only)?

Question

Please solve definite integral ∫x^2/(1+ e^sinx) dx from -2 to 2

A number means nothing to me. Show your complete solution or give some hints please.

Answer 1

ok I thought of a solution after 30 mins or so. This solution can solve this integral from -a to a for any number a, without finding the indefinite integral first.

you can make use of the fact that sin(-x) = -sin(x)

let f(x) = 1+e^sin(x).
f(-x) = 1+e^sin(-x) = 1+e^(-sin(x)).

Therefore f(x) = e^sin(x) f(-x).

Split the integral into 2 parts, one of them from 0 to a and the other from -a to 0. You have:

integral from 0 to a [x^2/(1+e^sin(x)] dx + integral -a to 0 [x^2/(1+e^sin(x)] dx

using the transformation described earlier to transform the 2nd part: (and note you can only do this because x^2 = (-x)^2)


integral from 0 to a [x^2/(1+e^sin(x)] dx + integral 0 to a [e^sin(x)*x^2/(1+e^sin(x)] dx

= integral from 0 to a [x^2(1+e^sin(x)) / (1+e^sin(x))] dx
= integral from 0 to a [x^2] dx
= a^3 / 3
Evaluating for a = 2 gives the answer of 8/3.

Answer 2

This function does not have a closed form.

I will use Simpson's Rule, with h = 0.02.

http://en.wikipedia.org/wiki/Simpson%27s_rule

It turns out the result is 8/3.
Use Excel,
... plug in the x-values
... plug in the y-values
... create a new array composed of 1,4,2,4,2,4,...,4,2,4,1
then get the sumproduct of the last two arrays.
Multiply that by h/3.
The answerer above reinforces the result I obtained.

§

Answer 3

you will possibly use integration by areas enable u=x and du=dx dv= e^(2x) dx and v= e^(2x)/2 applying the formulation it could be (xe^(2x))/2 - the mix of ((e^(2x)dx)/2) (xe^(2x))/2 - 0.25 intergal of e^(2x) dx Ans: (xe^(2x))/2 - 0.25 e^(2x) interior the 2nd section you are able to do away with the 0.5 and take it out of the crucial sign . then you definately can substitute the 2x for u and do the crucial with understand to u hence you will get yet another 0.5, and in case you're taking that 0.5 exterior the crucial you get 0.25 on the exterior!!! desire this helps!!!

Answer 4

answer = 8/3