find-the-value-of-0-1-x-1-4-1-x-3-4-x-dx-

Question Number 56188 by maxmathsup by imad last updated on 11/Mar/19

f i n d t h e v a l u e o f \int_{0}^{\infty} \frac{{(1 + x)}^{- \frac{1}{4}} - {(1 + x)}^{- \frac{3}{4}}}{x} d x

Answered by Smail last updated on 12/Mar/19

l e t t = \sqrt[4]{1 + x} \Rightarrow d x = 4 t^{3} d t

I = \int_{0}^{\infty} (\frac{1}{x \sqrt[4]{1 + x}} - \frac{1}{x \sqrt[4]{{(1 + x)}^{3}}}) d x

= 4 \int_{1}^{\infty} (\frac{t^{2}}{t^{4} - 1} - \frac{1}{t^{4} - 1}) d t = 4 \int_{1}^{\infty} (\frac{t^{2} - 1}{t^{4} - 1}) d t

= 4 \int_{1}^{\infty} \frac{d t}{t^{2} + 1} = 4 {[{t a n}^{- 1} (t)]}_{1}^{\infty} = 4 (\frac{π}{2} - \frac{π}{4})

= π

Commented by maxmathsup by imad last updated on 12/Mar/19

s i r s m a i l d o y o u s t u d y o r w o r k i n u s a ?

Commented by Smail last updated on 12/Mar/19

I d o b o t h f o r n o w .

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