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Find the value of \(\rm \int_{5}^{6} \frac{dx}{x^{2}-16}\)
1. \(\rm \frac{1}{4}log\frac{9}{5}\)
2. \(\rm \frac{1}{8}log\frac{9}{7}\)
3. \(\rm \frac{1}{8}log\frac{ 9}{5}\)
4. \(\rm \frac{1}{8}log\frac{7}{5}\)
5. None of these

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Correct Answer - Option 3 : \(\rm \frac{1}{8}log\frac{ 9}{5}\)

Concept:

  • \(\rm \int\frac{dx}{x^{2}-a^{2}}=\frac{1}{2a}log\frac{\left |x-a\right |}{\left |x+a\right |}+C\)
  • \(\rm \int_{0}^{x}{f(x) dx}=F(x)-F(0)\), where F(x) is the anti-derivative of f(x).


Calculation:

Given: \(\rm \int_{5}^{6} \frac{dx}{x^{2}-16}\)

Using the formula, \(\rm \int\frac{dx}{x^{2}-a^{2}}=\frac{1}{2a}log\frac{\left |x-a\right |}{\left |x+a\right |}+C\)

\(\rm \Rightarrow \int_{5}^{6} \frac{dx}{x^{2}-16}=\int_{5}^{6} \frac{dx}{x^{2}-4^{2}}=\left [ \frac{1}{8}log\frac{\left |x-4\right |}{\left |x+4\right |} \right ]_{5}^{6}\)

\(\rm\Rightarrow \int_{5}^{6} \frac{dx}{x^{2}-16}=\frac{1}{8}(log\frac{\left |6-4\right |}{\left |6+4\right |}-log\frac{\left |5-4\right |}{\left |5+4\right |})=\frac{1}{8}(log\frac{\left |2\right |}{\left |10\right |}-log\frac{\left |1\right |}{\left |9\right |})\)

It is known that log a - log b = log (a / b)

\(\rm\Rightarrow \int_{5}^{6} \frac{dx}{x^{2}-16}=\frac{1}{8}(log\frac{\left |2 \times9\right |}{\left |10\right |})=\frac{1}{8}log\frac{9}{5}\)

Hence, the correct answer is option 3.

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