Question

If the value of the integral

\[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left( \frac{x^2 \cos x}{1 + \pi^x} + \frac{1 + \sin^2 x}{1 + e^{\sin^x 2023}} \right) dx = \frac{\pi}{4} (\pi + a) - 2, \]

then the value of \(a\) is:

• A. 2
• B. \(-\frac{3}{2}\)
• C. 3
• D. \(\frac{3}{2}\)

Answer 1

The Correct Option is C

Step 1: Set Up the Integral \(I\)

\[ I = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left( \frac{x^2 \cos x}{1 + \pi^x} + \frac{1 + \sin^2 x}{1 + e^{\sin^{2023} x}} \right) dx \]

Step 2: Use Symmetry to Simplify

Notice that the integrand has symmetry properties, allowing us to split the integral and add:

\[ I = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left( \frac{x^2 \cos x}{1 + \pi^x} + \frac{1 + \sin^2 x}{1 + e^{\sin^{2023}(-x)}} \right) dx \]

Step 3: Combine Integrals

Adding the two integrals results in:

\[ 2I = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left( x^2 \cos x + 1 + \sin^2 x \right) dx \]

Step 4: Evaluate the Integral

Solving this integral gives:

\[ I = \frac{\pi^2}{4} + \frac{3\pi}{4} - 2 \]

Step 5: Determine \(a\)

From the given equation, equate terms to find \(a = 3\).

So, the correct answer is: \(a = 3\)