Evaluate the definite integral \int_{0}^{\frac{\pi}{2}} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} dx.

Get the complete, step-by-step math solution for: "Evaluate the definite integral \int_{0}^{\frac{\pi}{2}} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} dx....". Powered by SolveForX AI math tutor.

Step-by-Step Solution

Step 1: Define the integral

Let the given definite integral be denoted by $I$. This is the standard way to begin solving definite integral problems, especially when properties of definite integrals are to be applied.

Step 2: Apply the property of definite integrals

We apply the property of definite integrals which states that $\int_{a}^{b} f(x) dx = \int_{a}^{b} f(a+b-x) dx$. In this case, $a=0$ and $b=\frac{\pi}{2}$, so $a+b-x = \frac{\pi}{2}-x$. We substitute this into the integral.

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

We add equation (1) and equation (2) together. Since the limits of integration are the same, we can combine the integrands. Notice that the denominators are identical.

Step 5: Simplify and evaluate the integral

The numerator and denominator of the combined integrand are identical, so the fraction simplifies to $1$. We then integrate $1$ with respect to $x$, which gives $x$. Evaluating this from $0$ to $\frac{\pi}{2}$ yields $\frac{\pi}{2}$.

Step 6: Solve for I

Finally, we divide both sides by $2$ to find the value of $I$, which is $\frac{\pi}{4}$.