find-0-1-1-x-4-dx-

Question Number 51550 by maxmathsup by imad last updated on 28/Dec/18

f i n d \int_{0}^{1} \sqrt{1 + x^{4}} d x

Commented by Abdo msup. last updated on 29/Dec/18

l e t I = \int_{0}^{1} \sqrt{1 + x^{4}} d x n o e l e m e n t a r y f u n c t i o n s o

w e g i v e I a t f o r m o f s e r i e

w e h a v e {(1 + u)}^{α} = 1 + \frac{α u}{1!} + \frac{α (α - 1)}{2!} u^{2} +

\dots \frac{α (α - 1) \dots (α - n + 1)}{n!} u^{n} + \dots \Rightarrow

{(1 + x^{4})}^{\frac{1}{4}} = 1 + \frac{1}{4} x^{4} + \frac{1}{2} \frac{1}{4} (\frac{1}{4} - 1) x^{8} + \dots

\frac{\frac{1}{4} (\frac{1}{4} - 1) \dots (\frac{1}{4} - n + 1)}{n!} x^{4 n} + \dots

= 1 + \frac{x^{4}}{4} - \frac{3}{32} x^{8} + \dots \Rightarrow

I \sim \int_{0}^{1} (1 + \frac{x^{4}}{4} - \frac{3}{32} x^{8}) d x

= {[x + \frac{x^{5}}{20} - \frac{3}{9.32} x^{9}]}_{0}^{1} = 1 + \frac{1}{20} - \frac{1}{96}

= \frac{21}{20} - \frac{1}{96} .

Commented by Abdo msup. last updated on 29/Dec/18

I \sim 1, 04

Answered by tanmay.chaudhury50@gmail.com last updated on 28/Dec/18