Question

\( \int \sqrt{1 + \cos x} \, dx \) is equal to:

• A. \( 2 \sin \left( \frac{x}{2} \right) + C \)
• B. \( \sqrt{2} \sin \left( \frac{x}{2} \right) + C \)
• C. \( \frac{1}{2} \sin \left( \frac{x}{2} \right) + C \)
• D. \( \frac{\sqrt{2}}{2} \sin \left( \frac{x}{2} \right) + C \)
• E. \( 2\sqrt{2} \sin \left( \frac{x}{2} \right) + C \)

Hint:

Trigonometric identities are your best friend for integrals involving radicals. $1 + \cos x \to 2\cos^2(x/2)$ and $1 - \cos x \to 2\sin^2(x/2)$ are standard substitutions.

Answer 1

Answer: (E)

Concept: We use the half-angle trigonometric identity to simplify the square root: \( 1 + \cos x = 2 \cos^2 \left( \frac{x}{2} \right) \). This transforms the radical into a simple trigonometric function.

Step 1: Apply the identity.
\[ \int \sqrt{2 \cos^2 \left( \frac{x}{2} \right)} \, dx = \int \sqrt{2} \left| \cos \left( \frac{x}{2} \right) \right| \, dx \] Assuming the integration is within an interval where cosine is positive: \[ \sqrt{2} \int \cos \left( \frac{x}{2} \right) \, dx \]

Step 2: Integrate.
Using the basic integration rule \( \int \cos(ax) \, dx = \frac{\sin(ax)}{a} \), where \( a = 1/2 \): \[ \sqrt{2} \cdot \left[ \frac{\sin(x/2)}{1/2} \right] + C \] \[ = \sqrt{2} \cdot 2 \sin \left( \frac{x}{2} \right) + C \]

Step 3: Simplify.
\[ 2\sqrt{2} \sin \left( \frac{x}{2} \right) + C \]