dx-x-x-1-4-1-10-

Question Number 79612 by john santu last updated on 26/Jan/20

\int \frac{dx}{\sqrt{x} {(\sqrt[4]{x} + 1)}^{10}} = ?

Answered by MJS last updated on 26/Jan/20

\int \frac{d x}{x^{\frac{1}{2}} {(x^{\frac{1}{4}} + 1)}^{10}} =

[t = x^{\frac{1}{4}} \to d x = 4 x^{\frac{3}{4}} d t]

= 4 \int \frac{t}{{(t + 1)}^{10}} d t = 4 \int \frac{d t}{{(t + 1)}^{9}} - 4 \int \frac{d t}{{(t + 1)}^{10}} =

= - \frac{1}{2 {(t + 1)}^{8}} + \frac{4}{9 {(t + 1)}^{9}} = - \frac{9 t + 1}{18 {(t + 1)}^{9}} =

= - \frac{9 x^{\frac{1}{4}} + 1}{18 {(x^{\frac{1}{4}} + 1)}^{9}} + C

Commented by peter frank last updated on 26/Jan/20

t h a n k s

Commented by john santu last updated on 27/Jan/20

thank you mister