Question

$\int_{}^{}{((e/x)^{x} + (x/e)^{x})\mathcal{l}nxdx}$ =

• A. (x/e)^x – (e/x)^x + c
• B. (x/e)^x + (e/x)^x + c
• C. (e/x)^x – (x/e)^x + c
• D. (x/e)^x + 2 (e/x)^x + c

Answer 1

Answer: (A)

(x/e)^x – (e/x)^x + c

Answer 2

Let $\left( \frac{x}{e} \right)^{x}$ = t

$\left( \left( \frac { \mathrm { x } } { \mathrm { e } } \right) ^ { \mathrm { x } } \ln \frac { \mathrm { x } } { \mathrm { e } } + \mathrm { x } \cdot \left( \frac { \mathrm { x } } { \mathrm { e } } \right) ^ { \mathrm { x } - 1 } \cdot \frac { 1 } { \mathrm { e } } \right) \mathrm { dx }$ = dt

= dt ̃ $\left( \frac{x}{e} \right)^{x}\mathcal{l}nxdx$ =dt

So I = $\int_{}^{}{\left( \frac{1}{t^{2}} + 1 \right)dt}$ = $- \frac{1}{t}$+ t + C

= $\left( \frac { x } { e } \right) ^ { x } - \left( \frac { e } { x } \right) ^ { x } + C$