If   \int \frac{log\left ( t+\sqrt{1+t^{2}} \right )}{\sqrt{1+t^{2}}}\; dt=\frac{1}{2}\left ( g\left ( t \right ) \right )^{2}+C,    where C is a constant, then g(2) is equal to :

  • Option 1)

    2log\left ( 2+\sqrt{5} \right )

  • Option 2)

    log\left ( 2+\sqrt{5} \right )

  • Option 3)

    \frac{1}{\sqrt{5}}log\left ( 2+\sqrt{5} \right )

  • Option 4)

    \frac{1}{2}log\left ( 2+\sqrt{5} \right )

 

Answers (1)

As learnt in concept

Integration by substitution -

The functions when on substitution of the variable of integration to some quantity gives any one of standard formulas.

 

 

- wherein

Since \int f(x)dx=\int f(t)dt=\int f(\theta )d\theta all variables must be converted into single variable ,\left ( t\, or\ \theta \right )

 

 

=\int \frac{log(t+\sqrt{1+t^2})}{\sqrt{1+t^2}} dt

Put log(t+\sqrt{1+t^2})} =\:m

\frac{1}{t+\sqrt{1+t^2}}\times (1+\frac{2t}{2\sqrt{1+t^2}})dt=dm

\frac{dt}{\sqrt{1+t^2}}=dm

we\: get \:\int mdm =\frac{m^2}{2} +C

So g(t)=log(1+\sqrt{1+t^2})

Put \:t=2, \: we \: get \:g(2)= log (2+\sqrt{5}) 


Option 1)

2log\left ( 2+\sqrt{5} \right )

Incorrect

Option 2)

log\left ( 2+\sqrt{5} \right )

Correct

Option 3)

\frac{1}{\sqrt{5}}log\left ( 2+\sqrt{5} \right )

Incorrect

Option 4)

\frac{1}{2}log\left ( 2+\sqrt{5} \right )

Incorrect

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