let-f-t-0-3-t-x-x-2-dx-with-t-1-4-1-find-a-explicit-form-of-f-t-2-find-also-g-t-0-3-dx-t-x-x-2-3-calculate-0-3-1-x-x-2-dx-0-3-2-x-x-2-dx-0-

Question Number 60498 by abdo mathsup 649 cc last updated on 21/May/19

l e t f (t) = \int_{0}^{3} \sqrt{t + x + x^{2}} d x w i t h t ⩾ \frac{1}{4}

1) f i n d a e x p l i c i t f o r m o f f (t)

2) f i n d a l s o g (t) = \int_{0}^{3} \frac{d x}{\sqrt{t + x + x^{2}}}

3) c a l c u l a t e \int_{0}^{3} \sqrt{1 + x + x^{2}} d x, \int_{0}^{3} \sqrt{2 + x + x^{2}} d x

\int_{0}^{3} \frac{d x}{\sqrt{2 + x + x^{2}}} .

Commented by maxmathsup by imad last updated on 22/May/19

l e t x^{2} + x + t \to Δ = 1 - 4 t b u t 4 t ⩾ 1 \Rightarrow 4 t - 1 ⩾ 0 \Rightarrow 1 - 4 t ⩽ 0

c a s e 1 1 - 4 t < 0 \Rightarrow x^{2} + x + t = x^{2} + 2 \frac{x}{2} + \frac{1}{4} + t - \frac{1}{4} = {(x + \frac{1}{2})}^{2} + \frac{4 t - 1}{4}

w e d o t h e c h a n g e m e n t (x + \frac{1}{2}) = \frac{\sqrt{4 t - 1}}{2} u \Rightarrow u = \frac{2 x + 1}{\sqrt{4 t - 1}}

f (t) = \int_{\frac{1}{\sqrt{4 t - 1}}}^{\frac{7}{\sqrt{4 t - 1}}} \frac{\sqrt{4 t - 1}}{2} \sqrt{1 + u^{2}} \frac{\sqrt{4 t - 1}}{2} d u = \frac{4 t - 1}{4} \int_{\frac{1}{\sqrt{4 t - 1}}}^{\frac{7}{\sqrt{4 t - 1}}} \sqrt{1 + u^{2}} d u

=_{u = s h (α)} \frac{4 t - 1}{4} \int_{a r g s h (\frac{1}{\sqrt{4 t - 1}})}^{a r g s h (\frac{7}{\sqrt{4 t - 1}})} c h (α) c h (α) d α

= \frac{4 t - 1}{8} \int_{l n (\frac{1}{\sqrt{4 t - 1}} + \sqrt{1 + \frac{1}{4 t - 1}})}^{l n (\frac{7}{\sqrt{4 t - 1}} + \sqrt{1 + \frac{49}{4 t - 1}})} (1 + c h (2 t)) d t

= \frac{4 t - 1}{8} {l n (\frac{7}{\sqrt{4 t - 1}} + \sqrt{\frac{4 t + 48}{4 t - 1}}) - l n (\frac{1}{\sqrt{4 t - 1}} + \frac{2 \sqrt{t}}{\sqrt{4 t - 1}})}

+ \frac{4 t - 1}{16} {[s h (2 t)]}_{l n (\frac{1}{\sqrt{4 t} - 1} + \sqrt{\frac{4 t}{4 t - 1}})}^{l n (\frac{7}{\sqrt{4 t} - 1} + \sqrt{\frac{4 t + 48}{4 t - 1}})}

= \frac{4 t - 1}{8} {l n (\frac{7}{\sqrt{4 t - 1}} + \sqrt{\frac{4 t + 48}{4 t - 1}}) - l n (\frac{1}{\sqrt{4 t - 1}} + \sqrt{\frac{4 t}{4 t - 1}})}

\frac{4 t - 1}{32} {[e^{2 t} - e^{- 2 t}]}_{\dots}^{\dots} \Rightarrow

f (t) = \frac{4 t - 1}{8} {l n (\frac{7}{\sqrt{4 t - 1}} + \sqrt{\frac{4 t + 48}{4 t - 1}}) - l n (\frac{1}{\sqrt{4 t - 1}} + \sqrt{\frac{4 t}{4 g - 1}})}

+ \frac{4 t - 1}{32} {{(\frac{7}{\sqrt{4 t - 1}} + \sqrt{\frac{4 t + 48}{4 t - 1}})}^{2} - \frac{1}{{(\frac{7}{\sqrt{4 t - 1}} + \sqrt{\frac{4 t + 48}{4 t - 1}})}^{2}}

- {(\frac{1}{\sqrt{4 t - 1}} + \sqrt{\frac{4 t}{4 t - 1}})}^{2} + \frac{1}{{(\frac{1}{\sqrt{4 t - 1}} + \sqrt{\frac{4 t}{4 t - 1}})}^{2}}}

Commented by maxmathsup by imad last updated on 22/May/19

c a s e 2 t = \frac{1}{4} \Rightarrow f (t) = \int_{0}^{3} \sqrt{x^{2} + x + \frac{1}{4}} d x = \int_{0}^{3} \sqrt{{(x + \frac{1}{2})}^{2}} d x

= \int_{0}^{3} (x + \frac{1}{2}) d x = {[\frac{x^{2}}{2} + \frac{1}{2} x]}_{0}^{3} = \frac{9}{2} + \frac{3}{2} = 6

2) w e h a v e f^{^{'}} (t) = \int_{0}^{3} \frac{d x}{2 \sqrt{t + x + x^{2}}} = \frac{1}{2} g (t) \Rightarrow g (x) = 2 f^{^{'}} (t) r e s t t o {c a l c u l a t e f}^{^{'}} (t)

Commented by maxmathsup by imad last updated on 22/May/19

3) \int_{0}^{3} \sqrt{1 + x + x^{2}} d x = f (1) = \frac{3}{8} {l n (\frac{7}{\sqrt{3}} + \frac{\sqrt{52}}{\sqrt{3}}) - l n (\frac{1}{\sqrt{3}} + \frac{2}{\sqrt{3}})}

+ \frac{3}{32} {{(\frac{7}{\sqrt{3}} + \frac{\sqrt{52}}{\sqrt{3}})}^{2} - \frac{1}{{(\frac{7}{\sqrt{3}} + \frac{\sqrt{52}}{\sqrt{3}})}^{2}} - {(\frac{1}{\sqrt{3}} + \frac{2}{\sqrt{3}})}^{2} + \frac{1}{{(\frac{1}{\sqrt{3}} + \frac{2}{\sqrt{3}})}^{2}}} .

\int_{0}^{3} \sqrt{2 + x + x^{2}} d x = f (2) = \dots .

Commented by maxmathsup by imad last updated on 22/May/19

l e t c a l c u l a t e I = \int_{0}^{3} \frac{d x}{\sqrt{x^{2} + x + 2}} w e h a v e x^{2} + x + 2 = x^{2} + x + \frac{1}{4} + 2 - \frac{1}{4}

= {(x + \frac{1}{2})}^{2} + \frac{7}{4} l e t u s e t h e c h a n g e m e n t x + \frac{1}{2} = \frac{\sqrt{7}}{2} t a n θ \Rightarrow 2 x + 1 = \sqrt{7} t a n θ

I = \int_{a r c t a n (\frac{1}{\sqrt{7}})}^{a r c t a n (\sqrt{7})} \frac{1}{\sqrt{\frac{7}{4} (1 + {t a n}^{2} θ})} \frac{\sqrt{7}}{2} d θ = \int_{a r c t a n (\frac{1}{\sqrt{7}})}^{a r c t a n (\sqrt{7})} \frac{d θ}{\sqrt{1 + {t a n}^{2} θ}}

= \int_{a r c t a n (\frac{1}{\sqrt{7}})}^{a r c t a n (\sqrt{7})} c o s θ d θ = {[s i n θ]}_{a r c t a n (\frac{1}{\sqrt{7}})}^{a r c t a n (\sqrt{7})} = s i n (a r c t a n (\sqrt{7})) - s i n (a r c t a n (\frac{1}{\sqrt{7}}))

s i n (a r c t a n x) = \frac{x}{\sqrt{1 + x^{2}}} \Rightarrow s i n (a r c t a n (\sqrt{7})) = \frac{\sqrt{7}}{\sqrt{8}} = \frac{\sqrt{7}}{2 \sqrt{2}}

s i n (a r c t a n (\frac{1}{\sqrt{7}})) = \frac{1}{\sqrt{7} \sqrt{1 + \frac{1}{7}}} = \frac{1}{\sqrt{7} \frac{\sqrt{8}}{\sqrt{7}}} = \frac{1}{2 \sqrt{2}} \Rightarrow

I = \frac{\sqrt{7} - 1}{2 \sqrt{2}} .