# Find the value of $\rm \int_{5}^{6} \frac{dx}{x^{2}-16}$

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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.