Question

\(\int\limits_{2}^{8}\frac{5^{\sqrt{10-x}}}{5^{\sqrt{x}}+5^{\sqrt{10-x}}}\ dx=\)

• A. 4
• B. 5
• C. 3
• D. 6

Hint:

When dealing with integrals that have symmetric structures, try using substitution to exploit this symmetry. This can often simplify the integral and lead to a more straightforward solution, as seen in this example.

Answer 1

The Correct Option is C

The correct answer is (C) : 3.

Answer 2

The correct answer is: (C) 3.

We are asked to evaluate the integral:

\(\int\limits_{2}^{8}\frac{5^{\sqrt{10-x}}}{5^{\sqrt{x}}+5^{\sqrt{10-x}}}\ dx\)

Step 1: Analyze the integral structure

The given integral has a structure that suggests symmetry in the functions of \( x \) and \( 10 - x \). This symmetry can be exploited to simplify the integral. 
Step 2: Use symmetry and substitution

A common technique for handling integrals with such symmetry is to make a substitution. Let \( u = 10 - x \). This allows us to explore the behavior of the integrand under the transformation, which simplifies the calculation significantly. 
Step 3: Conclude the value of the integral

After performing the necessary steps using symmetry, we find that the value of the integral is 3. 
Therefore, the correct answer is (C) 3.