Determine whether the integral is convergent or divergent.81 cos nt dtO convergentO divergentIf it is convergent, evaluate it. (If the quantity diverges, enter DIVERGES.)

Given: ∫-∞∞81cosπtdt ∫-∞∞81cosπtdt=2∫0∞81cosπtdt=162∫0∞cosπtdt=162πlimb→∞sinπb-sin0=162πlimb→∞sinπb…

Answered: Determine whether the integral is…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Determine whether the integral is convergent or divergent.
81 cos nt dt
O convergent
O divergent
If it is convergent, evaluate it. (If the quantity diverges, enter DIVERGES.)

Expert Solution

Step 1

Given:

$\int_{- \infty}^{\infty} 81 \cos (πt) d t$

\int_{- \infty}^{\infty} 81 \cos (πt) d t = 2 \int_{0}^{\infty} 81 \cos (πt) d t = 162 \int_{0}^{\infty} \cos (πt) d t = \frac{162}{π} \lim_{b \to \infty} [\sin (πb) - \sin (0)] = \frac{162}{π} \lim_{b \to \infty} [\sin (πb)]

Since

\lim_{b \to \infty} [\sin (πb)]

does not exist (because sin(πb) oscillates between -1 and 1 forever), the given integral is DIVERGENT.

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Determine whether the integral is convergent or divergent. 81 cos nt dt O convergent O divergent If it is convergent, evaluate it. (If the quantity diverges, enter DIVERGES.)