1 2 e x ( 1 x - 1 x 2 ) ? d x =

Question

$\int_{1}^{2} e^{x} (\frac{1}{x} - \frac{1}{x^{2}}) d x =$

Smart Achievers · Accepted Answer

(3)

Ans. $\frac{e^{2}}{2} - e$

Explanation :

$\int_{1}^{2} e^{x} (\frac{1}{x} - \frac{1}{x^{2}}) d x = \int_{1}^{2} \frac{e^{x}}{x} d x - \int_{1}^{2} \frac{e^{x}}{x^{2}} d x$

$= [\frac{1}{x} \int_{1}^{2} e^{x} d x] - \int_{1}^{2} (\frac{- 1}{x^{2}} \int e^{x} d x) d x - \int_{1}^{2} \frac{1}{x^{2}} e^{x} d x$

$= {[\frac{1}{x} e^{x}]}_{1}^{2} + \int_{1}^{2} \frac{e^{x}}{x^{2}} d x - \int_{1}^{2} \frac{e^{x}}{x^{2}} d x$ $[∵ \int e^{x} {f (x) + f' (x)} d x = e^{x} f (x) + C]$

$= {[\frac{1}{x} e^{x}]}_{1}^{2} = \frac{e^{2}}{2} - e$

Solution

Q.
$\int_{1}^{2} e^{x} (\frac{1}{x} - \frac{1}{x^{2}}) d x =$

1 $\frac{e^{2}}{2} + e$

2 $e - \frac{e^{2}}{2}$

3 $\frac{e^{2}}{2} - e$

4 None of these

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Solution

Q. ∫12ex(1x-1x2) dx=

1 e22+e

2 e-e22

3 e22-e