Check the solution to "Evaluate the ff, improper integrals. ∫−2∞sin⁡xdx"

Khadija Wells

Answered question

2021-05-23

Evaluate the ff, improper integrals.

\int_{- 2}^{\infty} \sin x d x

Sally Cresswell

Skilled2021-05-24Added 91 answers

Evaluate the integral

\int_{- 2}^{\infty} \sin x d x = [\int \sin x d x]_{- 2}^{\infty}

= [- \cos (x)]_{- 2}^{\infty} = [- \cos (\infty) + \cos (- 2)] = [- \cos (\infty) + \cos 2]

Now, note that

lim_{x \to \infty} \cos (x)

doesn't exist, i.e. −

\cos (\infty)

doesn't exist.

\Rightarrow [- \cos (\infty) + \cos 2]

doesn't exist
Thus, the integral

\int_{- 2}^{\infty} \sin (x) d x

is divergent

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