calculate-1-6-1-x-1-x-2-x-dx-

Question Number 38714 by maxmathsup by imad last updated on 28/Jun/18

c a l c u l a t e \int_{1}^{6} \frac{{(- 1)}^{[x]}}{1 + x^{2} [x]} d x

Commented by abdo mathsup 649 cc last updated on 29/Jun/18

I = \sum_{k = 1}^{5} \int_{k}^{k + 1} \frac{{(- 1)}^{k}}{1 + {k x}^{2}} d x

= \sum_{k = 1}^{5} {(- 1)}^{k} \int_{k}^{k + 1} \frac{d x}{1 + {k x}^{2}} c h a n g e m e n t

x \sqrt{k} = t g i v e \int_{k}^{k + 1} \frac{d x}{1 + {k x}^{2}} = \int_{k \sqrt{k}}^{(k + 1) \sqrt{k}} \frac{1}{1 + t^{2}} \frac{d t}{\sqrt{k}} \Rightarrow

I = \sum_{k = 1}^{5} \frac{{(- 1)}^{k}}{\sqrt{k}} \int_{k \sqrt{k}}^{(k + 1) \sqrt{k}} \frac{d t}{1 + t^{2}}

= \sum_{k = 1}^{5} \frac{{(- 1)}^{k}}{\sqrt{k}} {a r c t a n (k + 1) \sqrt{k} - a r c t a n (k \sqrt{k})}

= - a r c t a n (2) + \frac{π}{4} + \frac{1}{\sqrt{2}} {a r c t a n (3 \sqrt{2}) - a r c t a n (2 \sqrt{2)}

- \frac{1}{\sqrt{3}} {a r c t a n (4 \sqrt{3}) - a r c t a n (3 \sqrt{3})}

+ \frac{1}{2} {a r 4 t a n (10) - a r c t a n (8)} - \frac{1}{\sqrt{5}} {a r c t a n 6 \sqrt{5)} - a r c t a n (5 \sqrt{5}) .

Answered by tanmay.chaudhury50@gmail.com last updated on 29/Jun/18

\int_{1}^{2} \frac{{(- 1)}^{1}}{1 + 1 \times x^{2}} + \int_{2}^{3} \frac{{(- 1)}^{2}}{1 + 2 x^{2}} + \int_{3}^{4} \frac{{(- 1)}^{3}}{1 + 3 x^{2}} + \int_{4}^{5} \frac{{(- 1)}^{4}}{1 + 4 x^{2}} +

\int_{5}^{6} \frac{{(- 1)}^{5}}{1 + 5 x^{2}}

= (- 1) ∣ {t a n}^{- 1} x ∣_{1}^{2} + \frac{1}{2} \times \frac{1}{\frac{1}{\sqrt{2}}} ∣ {t a n}^{- 1} (\frac{x}{\frac{1}{\sqrt{2}}}) ∣_{2}^{3} +

{(- 1)}^{} \times \frac{1}{3} \times \frac{1}{\frac{1}{\sqrt{3}}} ∣ {t a n}^{- 1} (\frac{x}{\frac{1}{\sqrt{3}}}) ∣_{3}^{4} +

\frac{1}{4} \times \frac{1}{\frac{1}{\sqrt{4}}} ∣ {t a n}^{- 1} (\frac{x}{\frac{1}{2}}) ∣_{4}^{5} + (- 1) \times \frac{1}{5} \times \frac{1}{\frac{1}{\sqrt{5}}} ∣ ({t a n}^{- 1} \frac{x}{\frac{1 \frac{}{}}{\sqrt{5}}}) ∣_{5}^{6}

= ({t a n}^{- 1} 1 - {t a n}^{- 1} 2) + \frac{1}{\sqrt{2}} ({t a n}^{- 1} 3 \sqrt{2} - {t a n}^{- 1} 2 \sqrt{2)}

- \frac{1}{\sqrt{3}} ({t a n}^{- 1} 4 \sqrt{3} - {t a n}^{- 1} 3 \sqrt{3}) + \frac{1}{2} ({t n}^{- 1} 10 - {t n}_{}^{- 1} 8

- \frac{1}{\sqrt{5}} ({t a n}^{- 1} 6 \sqrt{5} - {t a n}^{- 1} 5 \sqrt{5})

= {t a n}^{- 1} (\frac{- 1}{3}) + \frac{1}{\sqrt{2}} {t a n}^{- 1} (\frac{\sqrt{2}}{13}) - \frac{1}{\sqrt{3}} {t a n}^{- 1} (\frac{3 \sqrt{3}}{37})

+ \frac{1}{2} {t a n}^{- 1} (\frac{2}{81}) - \frac{1}{\sqrt{5}} {t a n}^{- 1} (\frac{\sqrt{5}}{151})