If-I-1-e-e-2-dx-log-x-and-I-2-1-2-e-x-x-dx-then-

Question Number 83077 by mhmd last updated on 27/Feb/20

If I_{1} = \underset{e}{\int^{e^{2}}} \frac{d x}{\log x} and I_{2} = \underset{1}{\int^{2}} \frac{e^{x}}{x} d x, then

Answered by TANMAY PANACEA last updated on 28/Feb/20

t = l n x \frac{d t}{d x} = \frac{1}{x} \to e^{t} d t = d x s o I_{1} = \int_{1}^{2} \frac{e^{t} d t}{t} = I_{2}

I_{1} = I_{2}

n o w e^{t} = 1 + t + \frac{t^{2}}{2!} + \frac{t^{3}}{3!} + \frac{t^{4}}{4!} + \dots

\int_{1}^{2} \frac{1 + t + \frac{t^{2}}{2!} + \frac{t^{3}}{3!} + \frac{t^{4}}{4!} + \dots}{t} d t

\int_{1}^{2} (\frac{1}{t} + 1 + \frac{t}{2!} + \frac{t^{2}}{3!} + \frac{t^{3}}{4!} + \dots) d t

∣ \frac{l n t}{1} + \frac{t}{1} + \frac{t^{2}}{2! \times 2} + \frac{t^{3}}{3! \times 3} + \dots ∣_{1}^{2}

= l n (\frac{2}{1}) + (2 - 1) + \frac{2^{2} - 1^{2}}{2! \times 2} + \frac{2^{3} - 1^{3}}{3! \times 3} + \dots