Question

Evaluate: \[ \int_{0}^{\frac{\pi}{4}} \frac{\sin x + \cos x}{9 + 16 \sin 2x} \, dx. \]

Hint:

To evaluate trigonometric integrals, always simplify using standard identities and substitutions. If the integral becomes too complex, consider numerical methods for final evaluation.

Answer 1

Step 1: Denominator and numerator simplification.

1. Denominator \(9 + 16 \sin 2x\): Apply \(\sin 2x = 2 \sin x \cos x\): \[ 9 + 16 \sin 2x = 9 + 16(2 \sin x \cos x) = 9 + 32 \sin x \cos x. \] 

2. Numerator \(\sin x + \cos x\): Apply the identity \(\sin x + \cos x = \sqrt{2} \sin\left(x + \frac{\pi}{4}\right)\). The integral transforms to: \[ \int_{0}^{\frac{\pi}{4}} \frac{\sqrt{2} \sin\left(x + \frac{\pi}{4}\right)}{9 + 32 \sin x \cos x} \, dx. \] 

Step 2: Substitution for simplification. Set \(\sin x = t\). 
This implies \(\cos x \, dx = dt\). The integration limits change as follows: 
Lower limit: \(x = 0 \implies \sin x = 0 \implies t = 0\). 
Upper limit: \(x = \frac{\pi}{4} \implies \sin x = \frac{\sqrt{2}}{2} \implies t = \frac{\sqrt{2}}{2}\). 

Using \(\sin x = t\), \(\cos x = \sqrt{1 - t^2}\), and \(\sin 2x = 2t\sqrt{1 - t^2}\), the integral becomes: \[ \int_{0}^{\frac{\sqrt{2}}{2}} \frac{\sqrt{2} \cdot \sin\left(\arcsin t + \frac{\pi}{4}\right)}{9 + 32 \cdot t \sqrt{1 - t^2}} \cdot \frac{dt}{\sqrt{1 - t^2}}. \] 

Step 3: Trigonometric term simplification. Use \(\sin(a + b) = \sin a \cos b + \cos a \sin b\): \[ \sin\left(\arcsin t + \frac{\pi}{4}\right) = t \cdot \frac{\sqrt{2}}{2} + \sqrt{1 - t^2} \cdot \frac{\sqrt{2}}{2}. \] 

Substitute this back into the integral: \[ \int_{0}^{\frac{\sqrt{2}}{2}} \frac{\sqrt{2} \left[t \cdot \frac{\sqrt{2}}{2} + \sqrt{1 - t^2} \cdot \frac{\sqrt{2}}{2}\right]}{9 + 32t\sqrt{1 - t^2}} \cdot \frac{dt}{\sqrt{1 - t^2}}. \] 

Further simplification and evaluation of this integral can be performed directly or via numerical methods. Final Answer: The exact evaluation is complex and may require advanced simplification or computational techniques. The simplified integrand facilitates numerical evaluation.