9. Evaluate the following integrals: (a) \( \int_{0}^{\pi / 4} \sqrt{1+\cos 4 x} d x \)

Calculate the integral \( \int_{0}^{\frac{\pi}{4}} \sqrt{1+\cos 4x} dx \). Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int_{0}^{\frac{\pi }{4}} \sqrt{1+\cos\left(4x\right)} dx\) - step1: Evaluate the power: \(\int_{0}^{\frac{\pi }{4}} \left(1+\cos\left(4x\right)\right)^{\frac{1}{2}} dx\) - step2: Simplify the expression: \(\int_{0}^{\frac{\pi }{4}} 2^{\frac{1}{2}}\left|\sin\left(2x\right)\right| dx\) - step3: Remove the absolute value bars: \(\int_{0}^{\frac{\pi }{4}} 2^{\frac{1}{2}}\sin\left(2x\right) dx\) - step4: Evaluate the integral: \(\int 2^{\frac{1}{2}}\sin\left(2x\right) dx\) - step5: Use properties of integrals: \(2^{\frac{1}{2}}\times \int \sin\left(2x\right) dx\) - step6: Evaluate the integral: \(2^{\frac{1}{2}}\left(-\frac{1}{2}\cos\left(2x\right)\right)\) - step7: Calculate: \(-\frac{2^{\frac{1}{2}}}{2}\cos\left(2x\right)\) - step8: Return the limits: \(\left(-\frac{2^{\frac{1}{2}}}{2}\cos\left(2x\right)\right)\bigg |_{0}^{\frac{\pi }{4}}\) - step9: Calculate the value: \(\frac{2^{\frac{1}{2}}}{2}\) - step10: Transform the expression: \(\frac{\sqrt{2}}{2}\) The value of the integral \( \int_{0}^{\frac{\pi}{4}} \sqrt{1+\cos 4x} dx \) is \( \frac{\sqrt{2}}{2} \).