let-f-x-2x-4x-dt-t-2-2t-3-1-find-f-x-2-calculate-lim-x-0-f-x-and-lim-x-f-x-

Question Number 57416 by Abdo msup. last updated on 03/Apr/19

l e t f (x) = \int_{2 x}^{4 x} \frac{d t}{t^{2} - 2 t + 3}

1) f i n d f (x)

2) c a l c u l a t e {l i m}_{x \to 0} f (x) a n d {l i m}_{x \to + \infty} f (x)

Commented by maxmathsup by imad last updated on 04/Apr/19

1) w e h a v e f (x) = \int_{2 x}^{4 x} \frac{d t}{t^{2} - 2 t + 1 + 2} = \int_{2 x}^{4 x} \frac{d t}{{(t - 1)}^{2} + 2} =_{t - 1 = \sqrt{2} u} \int_{\frac{2 x - 1}{\sqrt{2}}}^{\frac{4 x - 1}{\sqrt{2}}} \frac{\sqrt{2} d u}{2 (1 + u^{2})}

= \frac{1}{\sqrt{2}} {[a r c t a n u]}_{\frac{2 x - 1}{\sqrt{2}}}^{\frac{4 x - 1}{\sqrt{2}}} = \frac{1}{\sqrt{2}} {a r c t a n (\frac{4 x - 1}{\sqrt{2}}) - a r c t a n (\frac{2 x - 1}{\sqrt{2}})}

2) w e h a v e {l i m}_{x \to 0} f (x) = \frac{1}{\sqrt{2}} {- a r c t a n (\frac{1}{\sqrt{2}}) + a r c t a n (\frac{1}{\sqrt{2}})} = 0 a l s o

{l i m}_{x \to + \infty} f (x) = \frac{1}{\sqrt{2}} {\frac{π}{2} - \frac{π}{2}} = 0 .