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Evaluate\int \frac{dx}{\left ( x-1\right )\sqrt{x^{2}+x+1} }, x > 1.

  • Option 1)

    -\frac{1}{\sqrt{3}}ln\left | \frac{1}{x-1}-\frac{1}{2}+\sqrt{\frac{12\left ( \frac{1}{x-1} +\frac{1}{2}\right )^{2}+1}{12}} \right |+c

  • Option 2)

    \frac{1}{\sqrt{3}}ln\left | \frac{1}{x-1}-\frac{1}{2}+\sqrt{\frac{12\left ( \frac{1}{x-1} +\frac{1}{2}\right )^{2}+1}{12}} \right |+c

  • Option 3)

    \frac{1}{\sqrt{3}}ln\left | \frac{1}{x-1}+\frac{1}{2}+\sqrt{\frac{12\left ( \frac{1}{x-1} +\frac{1}{2}\right )^{2}+1}{12}} \right |+c

  • Option 4)

    \frac{1}{\sqrt{3}}ln\left | \frac{1}{x-1}-\frac{1}{2}+\sqrt{\frac{12\left ( \frac{1}{x-1} +\frac{1}{2}\right )^{2}+1}{12}} \right |+c

 

Answers (1)

best_answer

As we learnt

Type of Integration by perfect square -

The integration in the form 

(i) \int \frac{dx}{(px+q)\sqrt{ax^{2}+bx+c}}

(ii) \int \frac{dx}{(px+q)\sqrt{ax+b}}

(iii) \int \frac{(a+bx)^{m}}{(p+qx)^{n}}dx

- wherein

Working rule.

(i)  \rightarrow put  (px+q)=\frac{1}{t}

(ii) \rightarrow put  (ax+b)=t^{2}

(iii) \rightarrow put  (p+qx)=t

 

 Put x - 1 = 1/t and dx = -1/t2dt.

We{\rm{ }}\,get{\rm{ }}\,I\, = - \int {\frac{{dt}}{{\sqrt {3{t^2} + 3t + 1} }} = - \frac{1}{{\sqrt 3 }}\int {\frac{{dt}}{{\sqrt {{{\left( {t + \frac{1}{2}} \right)}^2} + \frac{1}{{12}}} }}} }

= - \frac{1}{{\sqrt 3 }}\ln \left| {\left( {t + 1/2} \right) + \sqrt {{{\left( {t + \frac{1}{2}} \right)}^2} + \frac{1}{{12}}} } \right| + c

= - \frac{1}{{\sqrt 3 }}\,\ln \,\left| {\frac{1}{{x - 1}} + \frac{1}{2} + \sqrt {\frac{{12{{\left( {\frac{1}{{x - 1}} + \frac{1}{2}} \right)}^2} + 1}}{{12}}} } \right| + c

 


Option 1)

-\frac{1}{\sqrt{3}}ln\left | \frac{1}{x-1}-\frac{1}{2}+\sqrt{\frac{12\left ( \frac{1}{x-1} +\frac{1}{2}\right )^{2}+1}{12}} \right |+c

Option 2)

\frac{1}{\sqrt{3}}ln\left | \frac{1}{x-1}-\frac{1}{2}+\sqrt{\frac{12\left ( \frac{1}{x-1} +\frac{1}{2}\right )^{2}+1}{12}} \right |+c

Option 3)

\frac{1}{\sqrt{3}}ln\left | \frac{1}{x-1}+\frac{1}{2}+\sqrt{\frac{12\left ( \frac{1}{x-1} +\frac{1}{2}\right )^{2}+1}{12}} \right |+c

Option 4)

\frac{1}{\sqrt{3}}ln\left | \frac{1}{x-1}-\frac{1}{2}+\sqrt{\frac{12\left ( \frac{1}{x-1} +\frac{1}{2}\right )^{2}+1}{12}} \right |+c

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gaurav

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