1-find-dx-x-2-5-x-3-9-2-calculate-4-dx-x-2-5-x-3-9-

Question Number 81719 by mathmax by abdo last updated on 14/Feb/20

1) f i n d \int \frac{d x}{{(x + 2)}^{5} {(x - 3)}^{9}}

2) c a l c u l a t e \int_{4}^{+ \infty} \frac{d x}{{(x + 2)}^{5} {(x - 3)}^{9}}

Commented by mathmax by abdo last updated on 15/Feb/20

l e t A = \int \frac{d x}{{(x + 2)}^{5} {(x - 3)}^{9}} \Rightarrow A = \int \frac{d x}{{(\frac{x + 2}{x - 3})}^{5} {(x - 3)}^{14}}

w e u s e t h e c h a n g e m e n t \frac{x + 2}{x - 3} = t \Rightarrow x + 2 = t x - 3 t \Rightarrow (1 - t) x = - 2 - 3 t

\Rightarrow (t - 1) x = 3 t + 2 \Rightarrow x = \frac{3 t + 2}{t - 1} \Rightarrow x - 3 = \frac{3 t + 2}{t - 1} - 3 = \frac{3 t + 2 - 3 t + 3}{t - 1}

= \frac{5}{t - 1} \Rightarrow d x = \frac{- 5}{{(t - 1)}^{2}} d t \Rightarrow

A = \int \frac{- 5}{{(t - 1)}^{2} t^{5} {(\frac{5}{t - 1})}^{14}} d t = \frac{- 5}{5^{14}} \int \frac{{(t - 1)}^{14}}{{(t - 1)}^{2} t^{5}} d t

= - \frac{1}{5^{13}} \int \frac{{(t - 1)}^{12}}{t^{5}} d t

= - \frac{1}{5^{13}} \int \frac{\sum_{k = 0}^{12} C_{12}^{k} t^{k} {(- 1)}^{12 - k}}{t^{5}} d t

= - \frac{1}{5^{13}} \int \frac{C_{12}^{0} - C_{12}^{1} t + C_{12}^{2} t^{2} - C_{12}^{3} t^{3} + C_{12}^{4} t^{4} - C_{12}^{5} t^{5} + C_{12}^{6} t^{6} - C_{12}^{7} t^{7} + C_{12}^{8} t^{8} - C_{12}^{9} t^{9} + C_{12}^{10} t^{10} - C_{12}^{11} t^{11} + C_{12}^{12} t^{12}}{t^{5}} d t

= - \frac{1}{5^{13}} \int (\frac{C_{12}^{0}}{t^{5}} - \frac{C_{12}^{1}}{t^{4}} + \frac{C_{12}^{2}}{t^{3}} - \frac{C_{12}^{3}}{t^{2}} + \frac{C_{12}^{4}}{t} - C_{12}^{5} + C_{12}^{6} t - C_{12}^{7} t^{2} + C_{12}^{8} t^{3} - C_{12}^{9} t^{4} + C_{12}^{10} t^{5} - C_{12}^{11} t^{6} + C_{12}^{12} t^{7}) d t

(- 5^{13}) A = - \frac{C_{12}^{0}}{4 t^{4}} + \frac{C_{12}^{1}}{3 t^{3}} - \frac{C_{12}^{2}}{2 t^{2}} + \frac{C_{12}^{3}}{t} + C_{12}^{4} l n ∣ t ∣ - C_{12}^{5} t + \frac{C_{12}^{6}}{2} t^{2} - \frac{C_{12}^{7}}{3} t^{3} + \frac{C_{12}^{8}}{4} t^{4} - \frac{C_{12}^{9}}{5} t^{5} + \frac{C_{12}^{10}}{6} t^{6} - \frac{C_{12}^{11}}{7} t^{7} + \frac{C_{12}^{12}}{8} t^{8} + C

(- 5^{13}) A = - \frac{1}{4} C_{12}^{0} {(\frac{x - 3}{x + 2})}^{4} + \frac{1}{3} C_{12}^{1} {(\frac{x - 3}{x + 2})}^{3} - \frac{1}{2} C_{12}^{2} {(\frac{x - 3}{x + 2})}^{2} + C_{12}^{3} (\frac{x - 3}{x + 2})

C_{12}^{4} l n ∣ \frac{x + 2}{x - 3} ∣ - C_{12}^{5} (\frac{x + 2}{x - 3}) + \frac{1}{2} C_{12}^{6} {(\frac{x + 2}{x - 3})}^{2} - \frac{1}{3} C_{12}^{7} {(\frac{x + 2}{x - 3})}^{3} + \frac{1}{4} C_{12}^{8} {(\frac{x + 2}{x - 3})}^{4}

- \frac{1}{5} C_{12}^{9} {(\frac{x + 2}{x - 3})}^{5} + \frac{1}{6} C_{12}^{10} {(\frac{x + 2}{x - 3})}^{6} - \frac{1}{7} C_{12}^{11} {(\frac{x + 2}{x - 3})}^{7} + \frac{1}{8} C_{12}^{12} {(\frac{x + 2}{x - 3})}^{8} + C

s o t h e v a l u e o f i n t e g r a l A i s k n o w n