calculate-1-2-dx-1-x-4-

Question Number 43675 by maxmathsup by imad last updated on 13/Sep/18

c a l c u l a t e \int_{1}^{2} \frac{d x}{1 + x^{4}} .

Commented by maxmathsup by imad last updated on 15/Sep/18

w e h a v e p r o v e d t h a t

\int \frac{d x}{1 + x^{4}} = \frac{1}{4 \sqrt{2}} l n ∣ \frac{x^{2} - \sqrt{2} x + 1}{x^{2} + \sqrt{2} x + 1} ∣ + \frac{1}{2 \sqrt{2}} {a r c t a n (x \sqrt{2} - 1) + a r c t a n (x \sqrt{2} + 1)} \Rightarrow

\int_{1}^{2} \frac{d x}{1 + x^{4}} = \frac{1}{4 \sqrt{2}} [l n ∣ \frac{x^{2} - \sqrt{2} x + 1}{x^{2} + \sqrt{2} x + 1} ∣_{1}^{2} + \frac{1}{2 \sqrt{2}} {[a r c t a n (x \sqrt{2} - 1) + a r c t a n (x \sqrt{2} + 1)]}_{1}^{2}

= \frac{1}{4 \sqrt{2}} {l n (\frac{5 - 2 \sqrt{2}}{5 + 2 \sqrt{2}}) - l n (\frac{2 - \sqrt{2}}{2 + \sqrt{2}})} + \frac{1}{2 \sqrt{2}} {a r c t a n (2 \sqrt{2} - 1) + a r c t a n (2 \sqrt{2} + 1)

- a r c t a n (\sqrt{2} - 1) - a r c t a n (\sqrt{2} + 1)} .