Examine the convergence of each of the following integrals. Where possible, prov

Question

Examine the convergence of each of the following integrals. Where possible, prove that the integral is absolutely convergent. If it is necessary to use Theorem VI, give details of the application of the theorem in the particular case. Proofs that an integral is not absolutely convergent need not be given. cos X dx. THEOREM V. If the integral Ja if(x) dx is convergent, so is Ja f(x) dx. In other words, if an integral is absolutely convergent, it is convergent. THEOREM VI. Consider an improper integral of first kind of the form where the functions and f satisfy the conditions: (a) d'(t) is continuous, db'(t)s0, and lim do (t) 0, t-on (b) f(t) is continuous, and the integral F(x)- f(t) dt is bounded for all x a. Then the integral (22.3-2) is convergent.

Explanation / Answer

let (t) = 1/(1-x^2)

then, '(t) = -2x/(1-x^2), which is less than 0

and lim t-->infinity 1/(1-x^2) = 1/infinity = 0

hence conditon a) is satified

now suppose F(x) = integral(cosx)dx from 0 to pi/2

= sin(pi/2) - sin0

= 1

this also satisfied the condition b)

hence the given fucntion is convergent

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