calculate-1-2-5-4-x-3-2-x-x-2-dx-

Question Number 42802 by maxmathsup by imad last updated on 02/Sep/18

c a l c u l a t e \int_{\frac{1}{2}}^{\frac{5}{4}} \frac{x^{3}}{\sqrt{2 + x - x^{2}}} d x

Commented by maxmathsup by imad last updated on 05/Sep/18

l e t A = \int_{\frac{1}{2}}^{\frac{5}{4}} \frac{x^{3}}{\sqrt{2 + x - x^{2}}} d x \Rightarrow A = \int_{\frac{1}{2}}^{\frac{5}{4}} \frac{x^{3}}{\sqrt{- x^{2} + x + 2}} d x

w e h a v e - x^{2} + x + 2 = - (x^{2} - x - 2) = - (x^{2} - 2 \frac{1}{2} x + \frac{1}{4} - \frac{1}{4} - 2)

= - {{(x - \frac{1}{2})}^{2} - \frac{9}{4}} = \frac{9}{4} - {(x - \frac{1}{2})}^{2} \Rightarrow A = \int_{\frac{1}{2}}^{\frac{5}{4}} \frac{x^{3}}{\sqrt{\frac{9}{4} - {(x - \frac{1}{2})}^{2}}} d x

=_{x - \frac{1}{2} = \frac{3}{2} s i n t} \int_{0}^{\frac{π}{6}} \frac{{(\frac{1}{2} + \frac{3}{2} s i n t)}^{3}}{\frac{3}{2} c o s t} \frac{3}{2} c o s t d t

= \int_{0}^{\frac{π}{6}} {(3 s i n t + 1)}^{3} d t = \int_{0}^{\frac{π}{6}} (27 {s i n}^{3} t + 27 {s i n}^{2} t + 9 s i n t + 1) d t

= 27 \int_{0}^{\frac{π}{6}} {s i n}^{3} t d t + 27 \int_{0}^{\frac{π}{6}} {s i n}^{2} t d t + 9 \int_{0}^{\frac{π}{6}} s i n t d t + \frac{π}{6} b u t

\int_{0}^{\frac{π}{6}} s i n t d t = {[- c o s t]}_{0}^{\frac{π}{6}} = 1 - \frac{\sqrt{3}}{2}

\int_{0}^{\frac{π}{6}} {s i n}^{2} t d t = \frac{1}{2} \int_{0}^{\frac{π}{6}} (1 - c o s (2 t)) d t = \frac{π}{12} - \frac{1}{4} {[s i n (2 t)]}_{0}^{\frac{π}{6}}

= \frac{π}{12} - \frac{1}{4} \frac{\sqrt{3}}{2} = \frac{π}{12} - \frac{\sqrt{3}}{8} w e h a v e {s i n}^{3} t = {\frac{e^{i t} - e^{- i t}}{2 i}}^{3}

= - \frac{1}{8 i} {\sum_{k = 0}^{3} C_{3}^{k} e^{i k t} {(- 1)}^{3 - k} e^{- i (3 - k) t}}

= \frac{i}{8} {- e^{- i 3 t} + 3 e^{i t} e^{- i 2 t} - 3 e^{i 2 t} e^{- i t} + e^{i 3 t}}

= \frac{i}{8} {e^{i 3 t} - e^{- i 3 t} - 3 (e^{i t} - e^{- i t})} = \frac{i}{8} {2 i s i n (3 t) - 6 i s i n (t)}

= - \frac{1}{4} s i n (3 t) + \frac{3}{4} s i n (t) \Rightarrow \int_{0}^{\frac{π}{6}} {s i n}^{3} t d t = \frac{3}{4} \int_{0}^{\frac{π}{6}} s i n (t) d t - \frac{1}{4} \int_{0}^{\frac{π}{6}} s i n (3 t) d t

= - \frac{3}{4} {[c o s t]}_{0}^{\frac{π}{6}} + \frac{1}{12} {[c o s (3 t)]}_{0}^{\frac{π}{6}} = - \frac{3}{4} (\frac{\sqrt{3}}{2} - 1) + \frac{1}{12} (- 1)

= \frac{3}{4} - \frac{3 \sqrt{3}}{8} - \frac{1}{12} s o t h e v a l u e o f A i s k n o w n .