calculate-0-e-x-2-e-x-x-dx-

Question Number 53601 by maxmathsup by imad last updated on 23/Jan/19

c a l c u l a t e \int_{0}^{\infty} \frac{e^{- x^{2}} - e^{- x}}{x} d x .

Answered by tanmay.chaudhury50@gmail.com last updated on 24/Jan/19

\int_{0}^{\infty} e^{- x^{2}} x^{- 1} d x - \int_{0}^{\infty} e^{- x} x^{- 1} d x

x = \sqrt{t} d x = \frac{1}{2 \sqrt{t}} d t

\int_{0}^{\infty} \frac{e^{- t} \times}{\sqrt{t}} \times \frac{d t}{2 \sqrt{t}} - \int_{0}^{\infty} e^{- x} \times \frac{d x}{x}

\frac{1}{2} \int_{0}^{\infty} e^{- t} \times \frac{d t}{t} - \int_{0}^{\infty} e^{- x} \times \frac{d x}{x}

g a m m a f u n c t i i n = \int_{0}^{\infty} e^{- x} x^{n - 1} d x = ⌈ (n)

⌈ (n + 1) = n ⌈ (n) = n! w h e n n > 0

s o w e c a n n o t u s e g a m m a f u n c t i o n . .

I_{1} = \int_{0}^{\infty} \frac{e^{- a x}}{x} d x

\frac{{d I}_{1}}{d a} = \int_{0}^{\infty} \frac{e^{- a x} \times - x}{x} d x = - \int_{0}^{\infty} e^{- a x} d x = - 1 \times ∣ \frac{e^{- a x}}{- a} ∣_{0}^{\infty}

\frac{{d I}_{1}}{d a} = \frac{1}{a} (\frac{1}{e^{\infty}} - \frac{1}{e^{0}} = - \frac{1}{a}

I_{1} = - l n a + c

w e h a v e t o f i n d v a l u e o f c w a i t \dots .