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