Question

If $\int {\dfrac{{\log (t + \sqrt {1 + {t^2}} )}}{{\sqrt {1 + {t^2}} }}} dt = \dfrac{1}{2}{(g(t))^2} + c$ where c is constant, then $g(2)$ is equal to,
A) $\dfrac{1}{{\sqrt 5 }}\log (2 + \sqrt 5 ){\text{ }}$
B) $2\log (2 + \sqrt 5 ){\text{ }}$
C) $\log (2 + \sqrt 5 ){\text{ }}$
D) $\dfrac{1}{2}\log (2 + \sqrt 5 ){\text{ }}$

Answer 1

Calculate the given integration using substitution method of the given function, we substitute $x = \log (t + \sqrt {1 + {t^2}} )$. And then on comparing both the sides find the value of $g\left( t \right)$. As the value of $g(2)$ can be calculated from there by replacing the value of t with two.

Complete step by step solution: As the given integration is of $\int {\dfrac{{\log (t + \sqrt {1 + {t^2}} )}}{{\sqrt {1 + {t^2}} }}} .dt$
So, by using substitution method,
$x = \log (t + \sqrt {1 + {t^2}} )$
Now, differentiate both sides and then substitute them in the above given equation as,

$$dx = \dfrac{1}{{t + \sqrt {1 + {t^2}} }}(1 + \dfrac{{2t}}{{2\sqrt {1 + {t^2}} }}).dt \\\ dx = \dfrac{1}{{t + \sqrt {1 + {t^2}} }}(1 + \dfrac{t}{{\sqrt {1 + {t^2}} }}).dt \\\$$

Now take L.C.M from the above equation and simplifying it as,

$$dx = \dfrac{1}{{t + \sqrt {1 + {t^2}} }}(\dfrac{{\sqrt {1 + {t^2}} }}{{\sqrt {1 + {t^2}} }} + \dfrac{t}{{\sqrt {1 + {t^2}} }}).dt \\\ dx = \dfrac{1}{{t + \sqrt {1 + {t^2}} }}(\dfrac{{\sqrt {1 + {t^2}} + t}}{{\sqrt {1 + {t^2}} }}).dt \\\$$

As cancelling the common terms, we get,
$dx = (\dfrac{1}{{\sqrt {1 + {t^2}} }}).dt$
Hence, replacing all the substitution in the above integration $\int {\dfrac{{\log (t + \sqrt {1 + {t^2}} )}}{{\sqrt {1 + {t^2}} }}} .dt$, we get,
$I = \int x .dx$
Hence, integrate the above term using general formula as,
$\Rightarrow I = \dfrac{{{x^2}}}{2} + c$
And on comparing it with R.H.S we can conclude that
$\Rightarrow I = \dfrac{{{x^2}}}{2} + c = \dfrac{1}{2}{(g(t))^2} + c$
So, the value of $g\left( t \right)$ is given as,
$\Rightarrow g(t) = x = \log (t + \sqrt {1 + {t^2}} )$
Hence, now calculate the value of $g(2)$ as replacing the value of t in the above equation,

$$\Rightarrow g(2) = \log (2 + \sqrt {1 + {2^2}} ) \\\ \Rightarrow g(2) = \log (2 + \sqrt 5 ) \\\$$

Hence, option (C) is our required correct answer.

Note: The substitution method is used when an integral contains some function and its derivative. In this case, we can set u equal to the function and rewrite the integral in terms of the new variable u. This makes the integral easier to solve.
Also apply the differentiation properly as $\dfrac{{d(\log x)}}{{dx}} = \dfrac{1}{x}$ and properly apply chain rule while further differentiation our given terms as $\dfrac{{d(\sqrt x )}}{{dx}} = \dfrac{1}{{2\sqrt x }}$.