Question Number 203681 by SANOGO last updated on 25/Jan/24
$$\int_{\mathrm{1}} ^{+{oo}} \frac{\mathrm{1}}{{x}\sqrt{{x}^{\mathrm{2}} +{x}+\mathrm{1}}}{dx} \\ $$
Answered by Frix last updated on 25/Jan/24
![t=((2x+1+2(√(x^2 +x+1)))/( (√3))) ⇒ ∫_(2+(√3)) ^∞ (1/(t−(√3)))−(3/(3t+(√3)))dt=[ln ((t−(√3))/(3t+(√3)))]_(2+(√3)) ^∞ = =ln (3+2(√3)) −ln 3](https://www.tinkutara.com/question/Q203684.png)
$${t}=\frac{\mathrm{2}{x}+\mathrm{1}+\mathrm{2}\sqrt{{x}^{\mathrm{2}} +{x}+\mathrm{1}}}{\:\sqrt{\mathrm{3}}} \\ $$$$\Rightarrow \\ $$$$\underset{\mathrm{2}+\sqrt{\mathrm{3}}} {\overset{\infty} {\int}}\frac{\mathrm{1}}{{t}−\sqrt{\mathrm{3}}}−\frac{\mathrm{3}}{\mathrm{3}{t}+\sqrt{\mathrm{3}}}{dt}=\left[\mathrm{ln}\:\frac{{t}−\sqrt{\mathrm{3}}}{\mathrm{3}{t}+\sqrt{\mathrm{3}}}\right]_{\mathrm{2}+\sqrt{\mathrm{3}}} ^{\infty} = \\ $$$$=\mathrm{ln}\:\left(\mathrm{3}+\mathrm{2}\sqrt{\mathrm{3}}\right)\:−\mathrm{ln}\:\mathrm{3} \\ $$
Answered by esmaeil last updated on 26/Jan/24
![∫^(+∞) _1 (dx/(x(√((x+(1/2))^2 +(3/4)))))= ^(∫_1 ^(+∞) ((2dx)/(x(√((2x+1)^2 +3))))=(2/( (√3)))∫_1 ^(+∞) (dx/( x(√((((2x+1)/( (√3))))^2 +1))))=Ω) ((2x+1)/( (√3)))=tanθ→dx=((√3)/(2cos^2 θ))dθ→ Ω=(2/( (√3)))×((√3)/2)∫_(π/3) ^(π/2) ((dθ/(cos^2 θ))/(tanθ×(1/(vosθ))))= =∫_(π/3) ^(π/2) cosecθdθ= ln∣cosecθ−cotθ∣]_(π/3) ^(π/2) = ln∣(1−((2(√3))/3)−((√3)/3))∣= ln((√3)−1) m](https://www.tinkutara.com/question/Q203707.png)
$$\underset{\mathrm{1}} {\int}^{+\infty} \frac{{dx}}{{x}\sqrt{\left({x}+\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} +\frac{\mathrm{3}}{\mathrm{4}}}}= \\ $$$$ \\ $$$$\:\:^{\int_{\mathrm{1}} ^{+\infty} \frac{\mathrm{2}{dx}}{{x}\sqrt{\left(\mathrm{2}{x}+\mathrm{1}\right)^{\mathrm{2}} +\mathrm{3}}}=\frac{\mathrm{2}}{\:\sqrt{\mathrm{3}}}\int_{\mathrm{1}} ^{+\infty} \frac{{dx}}{\:{x}\sqrt{\left(\frac{\mathrm{2}{x}+\mathrm{1}}{\:\sqrt{\mathrm{3}}}\overset{\mathrm{2}} {\right)}+\mathrm{1}}}=\Omega} \\ $$$$\frac{\mathrm{2}{x}+\mathrm{1}}{\:\sqrt{\mathrm{3}}}={tan}\theta\rightarrow{dx}=\frac{\sqrt{\mathrm{3}}}{\mathrm{2}{cos}^{\mathrm{2}} \theta}{d}\theta\rightarrow \\ $$$$\Omega=\frac{\mathrm{2}}{\:\sqrt{\mathrm{3}}}×\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\int_{\frac{\pi}{\mathrm{3}}} ^{\frac{\pi}{\mathrm{2}}} \frac{\frac{{d}\theta}{{cos}^{\mathrm{2}} \theta}}{{tan}\theta×\frac{\mathrm{1}}{{vos}\theta}}= \\ $$$$=\int_{\frac{\pi}{\mathrm{3}}} ^{\frac{\pi}{\mathrm{2}}} {cosec}\theta{d}\theta= \\ $$$$\left.{ln}\mid{cosec}\theta−{cot}\theta\mid\right]_{\frac{\pi}{\mathrm{3}}} ^{\frac{\pi}{\mathrm{2}}} = \\ $$$${ln}\mid\left(\mathrm{1}−\frac{\mathrm{2}\sqrt{\mathrm{3}}}{\mathrm{3}}−\frac{\sqrt{\mathrm{3}}}{\mathrm{3}}\right)\mid= \\ $$$${ln}\left(\sqrt{\mathrm{3}}−\mathrm{1}\right) \\ $$$${m} \\ $$