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प्रश्न
Evaluate:
`int(e^x)/(1 + e^-x) dx`
मूल्यांकन
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उत्तर
`int(e^x)/(1 + e^-x) dx`
= `int(e^x)/(1 + 1/e^x) dx`
= `int((e^x)^2)/(e^x + 1) dx`
= `int((e^x)^2 - 1 + 1)/(e^x + 1) dx`
= `int [((e^x)^2 - 1)/(e^x + 1) + 1/(e^x + 1)] dx`
= `int [((e^x - 1)(e^x + 1))/(e^x + 1) + (e^-x)/(1 + e^-x)] dx`
= `int [(e^x - 1) + (e^-x)/(1 + e^-x)] dx`
= `int e^x dx - int 1 dx - int (-e^-x)/(1 + e^-x) dx`
= `e^x - x - log |1 + e^-x| + c ...[because d/(dx) (1 + e^-x) = -e^-x and int (f'(x))/(f(x))dx = log f(x) + c]`
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