let-f-x-x-2-1-x-dt-1-t-t-2-1-calculate-f-x-interms-of-x-2-calculate-lim-x-0-f-x-and-lim-x-f-x-

Question Number 55273 by maxmathsup by imad last updated on 20/Feb/19

l e t f (x) = \int_{x^{2}}^{1 + x} \frac{d t}{1 + t + t^{2}}

1) c a l c u l a t e f (x) i n t e r m s o 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 21/Feb/19

1) w e h a v e f (x) = \int_{x^{2}}^{1 + x} \frac{d t}{t^{2} + 2 \frac{t}{2} + \frac{1}{4} + \frac{3}{4}} = \int_{x^{2}}^{1 + x} \frac{d t}{{(t + \frac{1}{2})}^{2} + \frac{3}{4}}

=_{t + \frac{1}{2} = \frac{\sqrt{3}}{2} u} \frac{4}{3} \int_{\frac{2 x^{2} + 1}{\sqrt{3}}}^{\frac{2 (1 + x) + 1}{\sqrt{3}}} \frac{1}{1 + u^{2}} \frac{\sqrt{3}}{2} d u = \frac{2}{\sqrt{3}} {[a r c t a n (u)]}_{\frac{2 x^{2} + 1}{\sqrt{3}}}^{\frac{2 x + 3}{\sqrt{3}}} \Rightarrow

f (x) = \frac{2}{\sqrt{3}} {a r c t a n (\frac{2 x + 3}{\sqrt{3}}) - a r c t a n (\frac{2 x^{2} + 1}{\sqrt{3}})}

2) {l i m}_{x \to 0} f (x) = \frac{2}{\sqrt{3}} {a r c t a n (\sqrt{3}) - a r c t a n (\frac{1}{\sqrt{3}})}

= \frac{2}{\sqrt{3}} {\frac{π}{3} - \frac{π}{6}} = \frac{2}{\sqrt{3}} {\frac{π}{6}} = \frac{π}{3 \sqrt{3}}

{l i m}_{x \to + \infty} f (x) = \frac{2}{\sqrt{3}} {a r c t a n (+ \infty) - a r c t a n (+ \infty)} = \frac{2}{\sqrt{3}} {\frac{π}{2} - \frac{π}{2}} = 0