let-f-x-0-cos-xt-x-2-t-2-dt-calculate-0-1-f-x-dx-

Question Number 131049 by mathmax by abdo last updated on 31/Jan/21

let f (x) = \int_{0}^{\infty} \frac{\cos (xt)}{x^{2} + t^{2}} dt calculate \int_{0}^{1} f (x) dx

Answered by mindispower last updated on 01/Feb/21

f (x) = \int_{0}^{\infty} \frac{c o s (x t)}{x^{2} + t^{2}} d t

t = x r \Rightarrow d t = x d r

= \frac{1}{x} \int_{0}^{\infty} \frac{c o s (r)}{1 + r^{2}} = \frac{1}{x} . \frac{1}{2} R e \int_{- \infty}^{\infty} \frac{e^{i r}}{1 + r^{2}} d r

= \frac{1}{2 x} R e s (\frac{e^{i r}}{1 + r^{2}}, r = i) = \frac{i π}{x}, \frac{e^{- 1}}{2 i} = \frac{π}{x e}

\int_{0}^{1} f (x) d x, n o t e x i s t