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Question Number 73181 by mathmax by abdo last updated on 07/Nov/19

c a l c u l a t e \int_{1}^{3} \frac{x - 2}{\sqrt{x^{2} + x + 1}} d x

Commented by mathmax by abdo last updated on 07/Nov/19

w e h a v e x^{2} + x + 1 = {(x + \frac{1}{2})}^{2} + \frac{3}{4} c h a n g e m e n t x + \frac{1}{2} = \frac{\sqrt{3}}{2} s h (t) g i v e

s h (t) = \frac{2 x + 1}{\sqrt{3}} \Rightarrow

\int_{1}^{3} \frac{x - 2}{\sqrt{x^{2} + x + 1}} d x = \int_{a r s h (\sqrt{3})}^{a r g s h (\frac{7}{\sqrt{3}})} \frac{\frac{\sqrt{3}}{2} s h (t) - \frac{1}{2} - 2}{\frac{\sqrt{3}}{2} c h (t)} \times \frac{\sqrt{3}}{2} c h (t) d t

= \int_{l n (\sqrt{3} + \sqrt{4})}^{l n (\frac{7}{\sqrt{3}} + \sqrt{1 + \frac{49}{3}})} (\frac{\sqrt{3}}{2} s h (t) - \frac{5}{2}) d t

= \frac{\sqrt{3}}{2} \int_{l n (\sqrt{3} + 2)}^{l n (\frac{7}{\sqrt{3}} + \sqrt{1 + \frac{49}{3}})} s h (t) d t - \frac{5}{2} (l n (\frac{7}{\sqrt{3}} + \frac{\sqrt{52}}{\sqrt{3}}) - l n (2 + \sqrt{3}))

= \frac{\sqrt{3}}{4} {[e^{t} + e^{- t}]}_{l n (2 + \sqrt{3})}^{l n (\frac{7 + \sqrt{52}}{\sqrt{3}})} - \frac{5}{2} {l n (\frac{7 + \sqrt{52}}{\sqrt{3}}) - l n (2 + \sqrt{3})}

= \frac{\sqrt{3}}{4} {(\frac{7 + \sqrt{52}}{\sqrt{3}}) + {(\frac{7 + \sqrt{52}}{\sqrt{3}})}^{- 1} - (2 + \sqrt{3}) - {(2 + \sqrt{3})}^{- 1}}

- \frac{5}{2} {l n (\frac{7 + \sqrt{52}}{\sqrt{3}}) - l n (2 + \sqrt{3})}

Answered by petrochengula last updated on 07/Nov/19

c o n s i d e r I = \int \frac{x - 2}{\sqrt{x^{2} + x + 1}} d x

= \frac{1}{2} \int \frac{2 (x - 2)}{\sqrt{x^{2} + x + 1}} d x = \frac{1}{2} \int \frac{2 x + 1}{\sqrt{x^{2} + x + 1}} d x - \frac{1}{2} \int \frac{5}{\sqrt{x^{2} + x + 1}} d x

= \sqrt{x^{2} + x + 1} - \frac{5}{2} \int \frac{1}{\sqrt{x^{2} + x + \frac{1}{4} + \frac{3}{4}}} d x

= \sqrt{x^{2} + x + 1} - \frac{5}{2} \int \frac{1}{\sqrt{{(x + \frac{1}{2})}^{2} + \frac{3}{4}}}

c o n s i d e r \int \frac{1}{\sqrt{{(x + \frac{1}{2})}^{2} + \frac{3}{4}}} d x

= \int \frac{1}{\sqrt{\frac{3}{4} (\frac{4}{3} {(\frac{2 x + 1}{2})}^{2} + 1)}} d x

= \frac{2}{\sqrt{3}} \int \frac{1}{\sqrt{{(\frac{2 x + 1}{\sqrt{3}})}^{2} + 1}} d x

l e t s i n h θ = \frac{2 x + 1}{\sqrt{3}} \Rightarrow \sqrt{3} s i n h θ = 2 x + 1 \Rightarrow \sqrt{3} c o s h θ d θ = 2 \frac{d x}{d θ} \Rightarrow d x = \frac{\sqrt{3}}{2} c o s h θ d θ = d x

= \int d θ

= θ + C

= {s i n h}^{- 1} (\frac{2 x + 1}{\sqrt{3}}) + c

\int \frac{x - 2}{\sqrt{x^{2} + x + 1}} d x = \sqrt{x^{2} + x + 1} - \frac{5}{2} {s i n h}^{- 1} (\frac{2 x + 1}{\sqrt{3}}) + C

\int_{1}^{3} \frac{x - 2}{\sqrt{x^{2} + x + 1}} d x = \sqrt{13} - \frac{5}{2} {s i n h}^{- 1} (\frac{7}{\sqrt{3}}) - \sqrt{3} + \frac{5}{2} {s i n h}^{- 1} (\frac{3}{\sqrt{3}})

Commented by petrochengula last updated on 07/Nov/19

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