2-dx-2-e-x-

Question Number 901 by 123456 last updated on 17/Apr/15

\underset{2}{\int^{+ \infty}} \frac{d x}{2 + e^{x}}

Answered by prakash jain last updated on 17/Apr/15

e^{x} = t

e^{x} d x = d t \Rightarrow d x = \frac{d t}{t}

\int \frac{d t}{t (2 + t)} = \frac{1}{2} [\int \frac{d t}{t} - \int \frac{d t}{2 + t}] = \frac{1}{2} \ln \frac{t}{2 + t} = \frac{1}{2} \ln \frac{e^{x}}{2 + e^{x}}

c a c u l a t i n g l i m i t s

I = - \frac{1}{2} \cdot \frac{e^{2}}{1 + e^{2}}