1. Definite Integral Property & Symmetry π ─── 2 √(sin x) Evaluate: ∫ ─────────────────── dx 0 √(sin x) + √(cos x)

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Step-by-Step Solution

Step 1: Define the integral

We begin by defining the given integral as $I$. This is a standard practice when solving definite integrals, especially when we anticipate using properties that might lead to a simpler form or a system of equations.

Step 2: Apply the property of definite integrals

We apply the property of definite integrals which states that $\int_{0}^{a} f(x) dx = \int_{0}^{a} f(a-x) dx$. Here, $a = \frac{\pi}{2}$. We substitute $x$ with $(\frac{\pi}{2}-x)$ in the integrand.

Step 3: Simplify the integral using trigonometric identities

Using the trigonometric identities $\sin (\frac{\pi}{2}-x) = \cos x$ and $\cos (\frac{\pi}{2}-x) = \sin x$, we simplify the integrand. This gives us a new expression for $I$, which we label as equation (2).

Step 4: Add the two integral equations

Now, we add equation (1) and equation (2). Since the limits of integration are the same and the denominators of the integrands are identical, we can combine them into a single integral.

Step 5: Simplify and integrate

The integrand simplifies to $1$. Integrating $1$ with respect to $x$ gives $x$. We then apply the limits of integration from $0$ to $\frac{\pi}{2}$.

Step 6: Solve for I

Finally, we solve for $I$ by dividing both sides of the equation by $2$. This gives us the value of the definite integral.