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If the given following function converges, give its value.

\int_{4}^{\infty} \frac{5 + sin(x)}{x}\ dx

asked Aug 19, 2017 by randomisation (1,920 points)  
edited Aug 19, 2017 by randomisation

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\text{Let}\ I = \int_{4}^{\infty} \frac{5 + sin(x)}{x} dx \\ = \int_{4}^{\infty} \frac{5}{x} dx + \int_{0}^{\infty} \frac{sin(x)}{x} dx - \int_{0}^{4} \frac{sin(x)}{x} dx \\ \\

Here, second integral is the Dirichlet integral [Can be easily solved by Laplace], thus equals, π/2

And the third integral is bounded which no singular points. So let it be k where k is a finite constant. Thus, 

I = 5\: log_{e} (x) |_ {4}^{\infty} + \frac{\pi}{2} - k \\ = 5 (log_{e}\infty - log_{e}4) + \frac{\pi}{2} - k \\ = \infty

Thus, clearly, the integral diverges, and hence, has no finite value.

answered Aug 19, 2017 by randomisation (1,920 points)  
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