calculate-P-1-e-pi-2-cos-lnx-x-dx-

Question Number 168429 by mathocean1 last updated on 10/Apr/22

c a l c u l a t e :

P = \int_{1}^{e^{\frac{π}{2}}} \frac{c o s (l n x)}{x} d x

Answered by qaz last updated on 10/Apr/22

\int_{1}^{e^{π / 2}} \frac{\cos (lnx)}{x} dx = ℜ \int_{1}^{e^{π / 2}} x^{i - 1} dx = ℜ \frac{x^{i}}{i} ∣_{1}^{e^{π / 2}} = ℜ (\frac{e^{\frac{π}{2} i}}{i} - \frac{1}{i}) = 1

Answered by FelipeLz last updated on 10/Apr/22

\ln (x) = u \to \frac{1}{x} d x = d u

P = \int_{1}^{e^{\frac{π}{2}}} \frac{\cos (\ln (x))}{x} d x = \int_{0}^{\frac{π}{2}} \cos (u) d u = \sin (u) ∣_{0}^{\frac{π}{2}} = 1

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