calculate-x-3-x-2-x-2-x-1-dx-

Question Number 62207 by maxmathsup by imad last updated on 17/Jun/19

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

Commented by maxmathsup by imad last updated on 18/Jun/19

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

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

H = \int \frac{\sqrt{3}}{2 \frac{\sqrt{3}}{2} \sqrt{1 + t^{2}}} d t = \int \frac{d t}{\sqrt{1 + t^{2}}} = l n (t + \sqrt{1 + t^{2}}) + c_{1} =

H = l n (\frac{2 x + 1}{\sqrt{3}} + \sqrt{1 + {(\frac{2 x + 1}{\sqrt{3}})}^{2}}) + c_{1}

K = \int \frac{1}{(\frac{\sqrt{3} t}{2} - \frac{1}{2} - 2) \frac{\sqrt{3}}{2} \sqrt{1 + t^{2}}} \frac{\sqrt{3}}{2} d t = \int \frac{d t}{(\frac{\sqrt{3}}{2} t - \frac{5}{2}) \sqrt{1 + t^{2}}} d t

= 2 \int \frac{d t}{(\sqrt{3} t - 5) \sqrt{1 + t^{2}}} =_{t = s h (u)} 2 \int \frac{c h (u) d u}{(\sqrt{3} s h (u) - 5) c h (u)}

= 2 \int \frac{d u}{\sqrt{3} s h (u) - 5} = 2 \int \frac{d u}{\sqrt{3} \frac{e^{u} - e^{- u}}{2} - 5} = 4 \int \frac{d u}{\sqrt{3} (e^{u} - e^{- u}) - 10}

=_{e^{u} = α} 4 \int \frac{1}{\sqrt{3} α - \sqrt{3} α^{- 1} - 10} \frac{d α}{α} = 4 \int \frac{d α}{\sqrt{3} α^{2} - \sqrt{3} - 10 α} = 4 \int \frac{d α}{\sqrt{3} α^{2} - 10 α - \sqrt{3}}

Δ^{^{'}} = {(- 5)}^{2} + 3 = 25 + 3 = 28 \Rightarrow α_{1} = \frac{5 + \sqrt{28}}{\sqrt{3}} a n d α_{2} = \frac{5 - \sqrt{28}}{\sqrt{3}} \Rightarrow

\int \frac{d α}{\sqrt{3} α^{2} - 10 α - \sqrt{3}} = \int \frac{d α}{\sqrt{3} (α - α_{1}) (α - α_{2})}

= \frac{1}{\sqrt{3} (2 \frac{\sqrt{28}}{\sqrt{3}})} \int (\frac{1}{α - α_{1}} - \frac{1}{α - α_{2}}) d α = \frac{1}{4 \sqrt{7}} l n ∣ \frac{α - α_{1}}{α - α_{2}} ∣ + c_{2} \Rightarrow

K = \frac{1}{\sqrt{7}} l n ∣ \frac{e^{u} - α_{1}}{e^{u} - α_{2}} ∣ + c_{2} b u t u = a r g s h (t) = l n (t + \sqrt{1 + t^{2}}) \Rightarrow e^{u} = t + \sqrt{1 + t^{2}}

K = \frac{1}{\sqrt{7}} l n ∣ \frac{t + \sqrt{1 + t^{2}} - α_{1}}{t + \sqrt{1 + t^{2} - α_{2}}} ∣ + c_{2} b u t t = \frac{2 x + 1}{\sqrt{3}} \Rightarrow

K = \frac{1}{\sqrt{7}} l n ∣ \frac{\frac{2 x + 1}{\sqrt{3}} + \sqrt{1 + {(\frac{2 x + 1}{\sqrt{3}})}^{2}} - α_{1}}{\frac{2 x + 1}{\sqrt{3}} + \sqrt{1 + {(\frac{2 x + 1}{\sqrt{3}})}^{2}} - α_{2}} ∣ + c_{2}

I = H + 5 K i s n o w n .

Answered by tanmay last updated on 17/Jun/19

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

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

I_{1} + 5 I_{2}

I_{1} = \int \frac{d x}{\sqrt{x^{2} + 2 \times x \times \frac{1}{2} + \frac{1}{4} + \frac{3}{4}}}

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

= l n [(x + \frac{1}{2}) + \sqrt{{(x + \frac{1}{2})}^{2} + {(\frac{\sqrt{3}}{2})}^{2}}] + c_{1}

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

x - 2 = \frac{1}{t} \to d x = \frac{- d t}{t^{2}}

\int \frac{- d t}{t^{2} \times \frac{1}{t} \times \sqrt{{(\frac{1}{t} + 2)}^{2} + (\frac{1}{t} + 2) + 1}}

\int \frac{- d t}{t \times \sqrt{\frac{1}{t^{2}} + \frac{4}{t} + 4 + \frac{1}{t} + 3}}

\int \frac{- d t}{t \times \sqrt{\frac{1 + 4 t + 4 t^{2} + t + 3 t^{2}}{t^{2}}}}

\int \frac{- d t}{\sqrt{7 t^{2} + 5 t + 1}}

\frac{1}{\sqrt{7}} \int \frac{- d t}{\sqrt{t^{2} + 2 \times t \times \frac{5}{14} + {(\frac{5}{14})}^{2} + \frac{1}{7} - \frac{25}{196}}}

\frac{1}{\sqrt{7}} \int \frac{- d t}{\sqrt{{(t + \frac{5}{14})}^{2} + \frac{28 - 25}{196}}}

\frac{1}{\sqrt{7}} \int \frac{- d t}{\sqrt{{(t + \frac{5}{14})}^{2} + \frac{3}{196}}}

\frac{- 1}{\sqrt{7}} \times l n [(t + \frac{5}{14}) + \sqrt{{(t + \frac{5}{14})}^{2} + \frac{3}{196}}]

= \frac{- 1}{\sqrt{7}} \times l n [(\frac{1}{x - 2} + \frac{5}{14}) + \sqrt{{(\frac{1}{x - 2} + \frac{5}{14})}^{2} + \frac{3}{196}})]

n o w p l s c a c u l a t e I_{1} + 5 I_{2}

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