calculate-lim-x-1-x-x-3-sh-xt-2-sin-xt-dt-

Question Number 78621 by mathmax by abdo last updated on 19/Jan/20

c a l c u l a t e {l i m}_{x \to 1} \int_{x}^{x^{3}} \frac{s h ({x t}^{2})}{s i n (x t)} d t

Commented by mathmax by abdo last updated on 22/Jan/20

l e t f (x) = \int_{x}^{x^{3}} \frac{s h ({x t}^{2})}{s i n (x t)} d t \Rightarrow f (x) =_{x t = u} \int_{x^{2}}^{x^{4}} \frac{s h (x \frac{u^{2}}{x^{2}})}{s i n (u)} \frac{d u}{x}

= \frac{1}{x} \int_{x^{2}}^{x^{4}} \frac{s h (\frac{1}{x} u^{2})}{s i n u} d u \exists c_{x} \in] x^{2}, x^{4} [/ f (x) = \frac{1}{x} s h (\frac{c_{x}^{2}}{x}) \int_{x^{2}}^{x^{4}} \frac{d u}{s i n u}

c h a n g e m e n t t a n (\frac{u}{2}) = z g i v e

\int_{x^{2}}^{x^{4}} \frac{d u}{s i n u} = \int_{t a n (\frac{x^{2}}{2})}^{t a n (\frac{x^{4}}{2})} \frac{1}{\frac{2 z}{1 + z^{2}}} \frac{2 d z}{1 + z^{2}} = \int_{t a n (\frac{x^{2}}{2})}^{t a n (\frac{x^{4}}{2})} \frac{d z}{z} = l n ∣ t a n (\frac{x^{4}}{2}) ∣ - l n ∣ t a n (\frac{x^{2}}{2}) ∣

\Rightarrow {l i m}_{x \to 1} f (x) = {l i m}_{x \to 1} \frac{1}{x} s h (\frac{c_{x}^{2}}{x}) l n ∣ \frac{t a n (\frac{x^{4}}{2})}{t a n (\frac{x^{2}}{2})} ∣

= s h (1) \times l n (1) = 0