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प्रश्न
Evaluate the integral by using substitution.
`int_1^2 (1/x- 1/(2x^2))e^(2x) dx`
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उत्तर
Let `I = int_1^2 e^(2x) (1/x - 1/(2x^2)) dx`
Put 2x = t
⇒ 2dx = dt
When x = 1, t = 2
And when x = 2, t = 4
∴ `I = 1/2 int_2^4 e^t (2/t - (1 xx4)/(2t^2)) dt`
`= 1/2 int_2^4 e^t (2/t - 2/t^2) dt`
`= int_2^4 e^t* (1/t - 1/t^2) dt`
`= int_2^4 e^t *[1/t + d/dt (1/t)] dt`
`= [e^t * 1/t]_2^4 = 1/4 e^4 - e^2/2`
`= e^2/2 (e^2/2 - 1)`
or `(e^2 (e^2 - 2))/4`
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