1) show that ∫_(−1) ^1 ((−dx)/( (√(x^2 +2x+2))))=ln((√5)−2) 2) determine ∫_1 ^∞ (dx/((x+1)(√x))) 3) test the convergence of the series given by Σ_(r=1) ^∞ (((r+1)!)/(r!(e^r ))) 4) obtain 3 non zero terms of maclaurrins series for sin^2 (x),hence evaluate ∫_0 ^(0.5) ((sin^2 (x))/x^2 )dx given your answer correct to 4 decimal places


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Question Number 174421 by ali009 last updated on 31/Jul/22

$1) s h o w t h a t$ $\int_{- 1}^{1} \frac{- d x}{\sqrt{x^{2} + 2 x + 2}} = l n (\sqrt{5} - 2)$ $2)$ $d e t e r m i n e$ $\int_{1}^{\infty} \frac{d x}{(x + 1) \sqrt{x}}$ $3) t e s t t h e c o n v e r g e n c e o f t h e s e r i e s g i v e n b y$ $\underset{r = 1}{\sum^{\infty}} \frac{(r + 1)!}{r! (e^{r})}$ $4)$ $o b t a i n 3 n o n z e r o t e r m s o f m a c l a u r r i n s$ $s e r i e s f o r {s i n}^{2} (x), h e n c e e v a l u a t e$ $\int_{0}^{0.5} \frac{{s i n}^{2} (x)}{x^{2}} d x$ $g i v e n y o u r a n s w e r c o r r e c t t o 4 d e c i m a l$ $p l a c e s$
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