find-the-value-of-0-dx-2x-1-2x-3-

Question Number 32360 by prof Abdo imad last updated on 23/Mar/18

f i n d t h e v a l u e o f \int_{0}^{\infty} \frac{d x}{(2 x + 1) (2 x + 3)} .

Commented by abdo imad last updated on 25/Mar/18

I = \int_{0}^{\infty} \frac{d x}{(2 x + 1) (2 x + 3)} = \frac{1}{2} \int_{0}^{\infty} (\frac{1}{2 x + 1} - \frac{1}{2 x + 3}) d x

\frac{1}{4} \int_{0}^{\infty} (\frac{2}{2 x + 1} - \frac{2}{2 x + 3}) d x = \frac{1}{4} {[l n ∣ \frac{2 x + 1}{2 x + 3} ∣]}_{0}^{\infty}

I = - \frac{1}{4} l n (\frac{1}{3}) = \frac{1}{4} l n (3) .