Question

Evaluate the integral (from -infinity to infinity) of cos(x)/(1+x^2).

Answer 1

Solve for M: integral (from 1 to M) 1/x^2 = 2

Answer 2

use substitution

Answer 3

this is a non-elementary integral. The best you con do is to estimation it.

-1 ≤ cosx ≤ 1

-1/(x^2+1) ≤ cosx/(x^2+1) ≤ 1/(x^2 + 1)

and so,
∫-1/(x^2+1)dx ≤ ∫ cosx/(x^2+1)dx ≤ ∫1/(x^2 + 1)

-pi ≤ ∫ cosx/(x^2+1)dx ≤ pi

Answer 4

ok, that was a very bad estimation XD

Answer 5

"Solve for M: integral (from 1 to M) 1/x^2 = 2"

\(\int \dfrac{1}{x^2}dx = -\dfrac{1}{x} + c \\ \dfrac{-1}{M}-\dfrac{-1}{1} = 2 \\ \dfrac{1}{M}=-2+1 \\ M = -1\)