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प्रश्न
Integrate the function:
`e^x/((1+e^x)(2+e^x))`
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उत्तर
Let `I = e^x/((1 + e^x)(2 + e^x))`
Put ex = t
ex dx = dt
∴ I = `int dt/((1 + t)(2 + t))`
Let `1/((1 + t)(2 + t)) = A/(1 + t) + B/(2 + t)`
`=> 1 = A(2 + t) + B(1 + t)` ....(1)
Putting t = -1 in equation (1),
∴ 1 = A(2 - 1)
⇒ A = 1
Putting t = -2 in equation (1),
∴ 1 = B(1 - 2)
⇒ B = - 1
`therefore 1/((1 + t)(2 + t)) dt`
`= int (1/(1 + t) - 1/(2 + t)) dt`
`therefore I = int 1/(1 + t) dt - int 1/(2 + t) dt`
`= log |1 + t| - log |2 + t| + C`
`= log |1 + e^x| - log |2 + e^x| + C`
`= log |(1 + t)/(2 + t)| + C`
`= log |(1 + e^x)/(2 + e^x)| + C`
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