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

Question Number 66338 by mathmax by abdo last updated on 12/Aug/19

f i n d \int_{\frac{1}{2}}^{\frac{5}{4}} \frac{x^{3} d x}{\sqrt{2 + x - x^{2}}}

Commented by mathmax by abdo last updated on 14/Aug/19

l e t I = \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 - \frac{1}{2})}^{2} - 2 - \frac{1}{4}) = - ({(x - \frac{1}{2})}^{2} - \frac{9}{4}) = \frac{9}{4} - {(x - \frac{1}{2})}^{2}

c h a n g e m e n t x - \frac{1}{2} = \frac{3}{2} t g i v e t = \frac{2 x - 1}{3} \Rightarrow

I = \int_{0}^{\frac{1}{2}} \frac{{(\frac{3}{2} t + \frac{1}{2})}^{3}}{\frac{3}{2} \sqrt{1 - t^{2}}} \frac{3}{2} d t = \frac{1}{8} \int_{0}^{\frac{1}{2}} \frac{{(3 t + 1)}^{3}}{\sqrt{1 - t^{2}}} d t

= \frac{1}{8} \int_{0}^{\frac{1}{2}} \frac{{(3 t)}^{3} + 3 {(3 t)}^{2} + 3 (3 t) + 1}{\sqrt{1 - t^{2}}} d t = \frac{1}{8} \int_{0}^{\frac{1}{2}} \frac{27 t^{3} + 27 t^{2} + 9 t + 1}{\sqrt{1 - t^{2}}} d t

= \frac{27}{8} \int_{0}^{\frac{1}{2}} \frac{t^{3}}{\sqrt{1 - t^{2}}} d t + \frac{27}{8} \int_{0}^{\frac{1}{2}} \frac{t^{2}}{\sqrt{1 - t^{2}}} d t + \frac{9}{8} \int_{0}^{\frac{1}{2}} \frac{t}{\sqrt{1 - t^{2}}} d t + \frac{1}{8} \int_{0}^{\frac{1}{2}} \frac{d t}{\sqrt{1 - t^{2}}}

w e h a v e \int_{0}^{\frac{1}{2}} \frac{t^{3}}{\sqrt{1 - t^{2}}} d t =_{t = s i n α} \int_{0}^{\frac{π}{6}} \frac{{s i n}^{3} α}{c o s α} c o s α d α

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

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

\int_{0}^{\frac{1}{2}} \frac{t^{2}}{\sqrt{1 - t^{2}}} d t =_{t = s i n κ} \int_{0}^{\frac{π}{6}} \frac{{s i n}^{2} α}{c o s α} c o s α d α = \int_{0}^{\frac{π}{6}} \frac{1 - c o s (2 α)}{2} d α

= \frac{π}{12} - \frac{1}{4} {[s i n (2 α)]}_{0}^{\frac{π}{6}} = \frac{π}{12} - \frac{1}{4} (\frac{\sqrt{3}}{2}) = \frac{π}{12} - \frac{\sqrt{3}}{8}

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

= 1 - \frac{\sqrt{3}}{2}

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

I = \frac{27}{8} {1 - \frac{\sqrt{3}}{2} + \frac{1}{3} (\frac{3 \sqrt{3}}{8} - 1)} + \frac{27}{8} {\frac{π}{12} - \frac{\sqrt{3}}{8}} + \frac{9}{8} (1 - \frac{\sqrt{3}}{2}) + \frac{π}{48}