Convergence-of-1-I-1-e-t-5-sin-lnt-t-1-3-2-dt-2-I-1-lnx-x-1-x-dx-

Question Number 79325 by Henri Boucatchou last updated on 24/Jan/20

C o n v e r g e n c e o f :

1) I = \int_{1}^{\infty} \frac{e^{- t / 5} ∣ s i n (l n t) ∣}{{(t - 1)}^{3 / 2}} d t

2) I = \int_{1}^{\infty} \frac{\sqrt{l n x}}{(x - 1) \sqrt{x}} d x

Commented by mathmax by abdo last updated on 24/Jan/20

1) c h a n g e m e n t t - 1 = x g i v e

I = \int_{1}^{+ \infty} \frac{e^{- \frac{t}{5}} ∣ s i n (l n t) ∣}{{(t - 1)}^{\frac{3}{2}}} d t = \int_{0}^{\infty} \frac{e^{- \frac{1 + x}{5}} ∣ s i n (l n (1 + x) ∣}{x^{\frac{3}{2}}} d x = \int_{0}^{\infty} φ (x) d x

I = \int_{0}^{1} (\dots) d x + \int_{1}^{+ \infty} (\dots) d x a t V (0) ∣ s i n (l n (1 + x) ∣\sim∣ s i n (x) ∣\sim∣ x ∣

\Rightarrow φ (x) \sim \frac{e^{- \frac{1}{5}} x}{x^{\frac{3}{2}}} = \frac{e^{- \frac{1}{5}}}{x^{\frac{1}{2}}} a n d \int_{0}^{1} e^{- \frac{1}{5}} x^{- \frac{1}{2}} d x c o n v e r g e s

a t + \infty ∣ φ (x) ∣ ⩽ \frac{e^{- \frac{1 + x}{5}}}{x^{\frac{3}{2}}} \Rightarrow \int_{1}^{+ \infty} ∣ φ (x) ∣ d x ⩽ \int_{1}^{+ \infty} x^{- \frac{3}{2}} e^{- \frac{1 + x}{5}} d x

w e h a v e l i m x^{2} x^{- \frac{3}{2}} e^{- \frac{1 + x}{5}} = 0 \Rightarrow t h i s i n t e g r a l c o n v e r g e s \Rightarrow

I c o n v e r g e s

Commented by mathmax by abdo last updated on 24/Jan/20

2) I = \int_{1}^{+ \infty} \frac{\sqrt{l n x}}{(x - 1) \sqrt{x}} d x c h a n g e m e n t \sqrt{l n x} = t g i v e l n x = t^{2} \Rightarrow

x = e^{t^{2}} \Rightarrow I = \int_{0}^{\infty} \frac{t}{(e^{t^{2}} - 1) e^{\frac{t^{2}}{2}}} \times 2 t e^{t^{2}} d t = \int_{0}^{\infty} \frac{2 t^{2} e^{\frac{t^{2}}{2}}}{e^{t^{2}} - 1} d t

=_{} \int_{0}^{\infty} \frac{2 t^{2} e^{- t^{2}} e^{\frac{t^{2}}{2}}}{1 - e^{- t^{2}}} d t = \int_{0}^{\infty} \frac{2 t^{2} e^{- \frac{t^{2}}{2}}}{1 - e^{- t^{2}}} d t = \int_{0}^{1} (\dots) d t + \int_{1}^{+ \infty} (\dots) d t

a t V (0) e^{- t^{2}} \sim 1 - t^{2} \Rightarrow 1 - e^{- t^{2}} \sim 1 - 1 + t^{2} \Rightarrow \Rightarrow \frac{2 t^{2} e^{- \frac{t^{2}}{2}}}{1 - e^{- t^{2}}} \sim 2 e^{- \frac{t^{2}}{2}} \to 2 (t \to 0)

\Rightarrow \int_{0}^{1} \frac{2 t e^{- \frac{t^{2}}{2}}}{1 - e^{- t^{2}}} d t c o n v e r g e s l e t ξ > 0 w e h a v e

{l i m}_{t \to + \infty} t^{1 + ξ} \times \frac{2 t e^{- \frac{t^{2}}{2}}}{1 - e^{- t^{2}}} = 0 \Rightarrow \int_{1}^{+ \infty} \frac{2 t^{2} e^{- \frac{t^{2}}{2}}}{1 - e^{- t^{2}}} c o n v e r g e s

f i n a l l y I i s c o n v e r g e n t