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

Question Number 123710 by Bird last updated on 27/Nov/20

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

Answered by MJS_new last updated on 27/Nov/20

Ostrogradski gives

\int \frac{d x}{{(x^{2} + 1)}^{2} {(x + 2)}^{5}} =

= - \frac{372 x^{5} + 2502 x^{4} + 6016 x^{3} + 6383 x^{2} + 4144 x + 3131}{7500 (x^{2} + 1) {(x + 2)}^{4}} -

- \frac{1}{625} \int \frac{31 x + 17}{(x^{2} + 1) (x + 2)} d x

- \frac{1}{625} \int \frac{31 x + 17}{(x^{2} + 1) (x + 2)} d x =

= - \frac{79}{6250} \int \frac{2 x}{x^{2} + 1} d x + \frac{3}{3125} \int \frac{d x}{x^{2} + 1} + \frac{79}{3125} \int \frac{d x}{x + 2} =

= - \frac{79}{6250} \ln (x^{2} + 1) + \frac{3}{3125} \arctan x + \frac{79}{3125} \ln ∣ x + 2 ∣ + C

\Rightarrow answer is

\frac{π}{12500} - \frac{146149}{81000} + \frac{2609 \sqrt{3}}{2500} + \frac{79}{6250} \ln \frac{7 + 4 \sqrt{3}}{18}

Commented by mathmax by abdo last updated on 28/Nov/20

thank you sir mjs

Answered by mathmax by abdo last updated on 28/Nov/20

complex method

I = \int_{1}^{\sqrt{3}} \frac{dx}{{(x - i)}^{2} {(x + i)}^{2} {(x + 2)}^{5}} = \int_{1}^{\sqrt{3}} \frac{dx}{{(\frac{x - i}{x + i})}^{2} {(x + i)}^{4} {(x + 2)}^{5}}

we do the changement \frac{x - i}{x + i} = t \Rightarrow x - i = tx + it \Rightarrow (1 - t) x = it + i \Rightarrow

x = i \frac{1 + t}{1 - t} \Rightarrow \frac{dx}{dt} = i \frac{1 - t - (1 + t) (- 1)}{{(1 - t)}^{2}} = i \frac{1 - t + 1 + t}{{(1 - t)}^{2}} = \frac{2 i}{{(1 - t)}^{2}}

x + i = \frac{i + it}{1 - t} + i = \frac{i + it + i - it}{1 - t} = \frac{2 i}{1 - t}

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

I = \int_{\frac{1 - i}{1 + i}}^{\frac{\sqrt{3} - i}{\sqrt{3} + i}} \frac{2 i}{{(1 - t)}^{2} t^{2} \frac{{(2 i)}^{4}}{{(1 - t)}^{4}} \frac{{((- 2 + i) t + 2 + i)}^{5}}{{(1 - t)}^{5}}} dt

= \frac{- 1}{{(2 i)}^{3}} \int_{\frac{1 - i}{1 + i}}^{\frac{\sqrt{3} - i}{\sqrt{3} + i}} \frac{{(t - 1)}^{9}}{{(t - 1)}^{2} t^{2} {(- 2 + i) t + 2 + i}^{5}} dt

= - \frac{i}{8} \int_{\frac{1 - i}{1 + i}}^{\frac{\sqrt{3} - i}{\sqrt{3} + i}} \frac{{(t - 1)}^{7}}{t^{2} {(- 2 + i) t + 2 + i}^{5}} dt

= \frac{i}{8} \int_{\frac{1 - i}{1 + i}}^{\frac{\sqrt{3} - i}{\sqrt{3} + i}} \frac{\sum_{k = 0}^{7} {(- 1)}^{k} C_{7}^{k} t^{k} {(- 1)}^{7 - k}}{t^{2} {(- 2 + i) t + 2 + i}^{5}} dt

rst decomposition \dots . be continued \dots