Question

what's the right answer ​

Answer 1

Given :

• Evaluate $\sf{∫ x^{-11} ( 1 + x^{4})^-½\:dx}$

To find :

• Evaluate $\sf{∫ x^{-11} ( 1 + x^{4})^-½\:dx}$

According to the question :

Here $\sf\frac{m + 1}{n} + p$ = $\sf\frac{[ -11 + 1}{4}]$ $\sf[ \frac{1}{2}]$ = -3

We are substituting ( 1 + x⁴ ) = t² x⁴,

then 1 + $\sf\frac{1}{x⁴}$ = t² and $\sf\frac{-4}{x^5}$ DC = 2t dt

⟹ $I = \sf{∫( \frac{dx}{x^11 ( 1 + x^4 )})^½}$

⟹ $I = \sf{∫( \frac{dx}{x^{11} × x² ( 1 + ( \frac{1}{x^4})})^½}$

⟹ $I = \sf{∫( \frac{dx}{x^{13} ( 1 + ( \frac{1}{x^4})})^½}$

⟹ $\sf{-¼ ∫( \frac{2t\:dt}{x^8\:t}})$

⟹ $\sf{-½ ∫( t² - t )²\:dt}$

⟹ $\sf{-½ ∫( t^{4} - 2t² + 1 )\:dt}$

⟹ $\sf{-½ ∫[( \frac{t^{5}}{5}) - ( \frac{2t³}{3}) + \:t}\:] +\:c$

⟹ $\pink \sf{t = \sqrt{1 + ( \frac{1}{x^4})}}$

So, It's Done !!