evaluate...evaluate; integral -10 to 10 (2e^x)/(sinhx+coshx) dx

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sciencesolve eNotes educator| Certified Educator

You need to use the following formulas for `sinh x`  and `cosh x`  such that:

`sinh x = (e^x - e^(-x))/2`

`cosh x = (e^x+e^(-x))/2`

`sinh x + cosh x = (e^x - e^(-x) + e^x+ e^(-x))/2`

Reducing like terms yields:

`sinh x + cosh x = (2e^x)/2 => sinh x + cosh x = e^x`

Substituting `e^x`  for `sinh x + cosh x`  yields:

`int_(-10)^10 (2e^x)/e^x dx = int_(-10)^10 2 dx`

`int_(-10)^10 2 dx = 2x|_(-10)^10`

`int_(-10)^10 2 dx = 2(10 - (-10)) = 2*20`

`int_(-10)^10 2 dx = 40`

Hence, evaluating the given definite integral yields `int_(-10)^10 (2e^x)/(sinh x + cosh x) dx = 40.`

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