Correct Answer - Option 2 : 54
Concept:
- \(\rm \int\frac{dx}{\sqrt {x^{2}-a^{2}}}=log{\left |x+\sqrt {x^{2}-a^{2}}\right |}+C\)
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\(\rm \int_{0}^{x}{f(x) dx}=F(x)-F(0)\), where F(x) is the anti-derivative of f(x).
Calculation:
Given: \(\rm \int_{4}^{6} \frac{dx}{\sqrt {x^{2}-4}}=log\frac{ {|a+\sqrt {b}}|}{ {|c+\sqrt {d}}|}\)
Using the formula, \(\rm \int\frac{dx}{\sqrt {x^{2}-a^{2}}}=log{\left |x+\sqrt {x^{2}-a^{2}}\right |}+C\)
\(⇒ \rm \int_{4}^{6} \frac{dx}{\sqrt {x^{2}-4}}=\left [ log{\left |x+\sqrt {x^{2}-4}\right |} \right ]_{4}^{6}\)
\(⇒ \rm \int_{4}^{6} \frac{dx}{\sqrt {x^{2}-4}}=log{|6+\sqrt {6^{2}-4}}|-log{|4+\sqrt {4^{2}-4}}|=log{|6+\sqrt {32}}|-log{|4+\sqrt {12}}|\)
It is known that log a - log b = log (a / b)
\(⇒ \rm \int_{4}^{6} \frac{dx}{\sqrt {x^{2}-4}}=log\frac{ {|6+\sqrt {32}}|}{ {|4+\sqrt {12}}|}\) ---(1)
∵ It is given that \(\rm \int_{4}^{6} \frac{dx}{\sqrt {x^{2}-4}}=log\frac{ {|a+\sqrt {b}}|}{ {|c+\sqrt {d}}|}\) ---(2)
By comparing (1) and (2), we get a = 6, b = 32, c = 4, d = 12.
⇒ a + b + c + d = 6 + 32 + 4 + 12 = 54
Hence, the correct answer is option 2.
