π ─── 2 √(sin x) Evaluate: ∫ ─────────────────── dx 0 √(sin x) + √(cos x)

Get the complete, step-by-step math solution for: "π ─── 2 √(sin x) Evaluate: ∫ ─────────────────── dx 0 √(sin x) + √(cos x)...". Powered by SolveForX AI math tutor.

Step-by-Step Solution

Step 1: Define the integral

Let the given integral be denoted by $I$. This is the standard way to begin solving definite integrals, especially when a property might be applied.

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 know that $\sin(\frac{\pi}{2} - x) = \cos x$ and $\cos(\frac{\pi}{2} - x) = \sin x$.

Step 3: Simplify the integral using trigonometric identities

By substituting the trigonometric identities $\sin(\frac{\pi}{2} - x) = \cos x$ and $\cos(\frac{\pi}{2} - x) = \sin x$ into the integral from the previous step, we get a new form of the integral.

Step 4: Add the original and transformed integrals

Now, we add the original integral (1) and the transformed integral (2). Since both integrals have the same limits and a common denominator, we can combine their integrands.

Step 5: Simplify the integrand

The numerator and denominator of the combined integrand are identical, so the fraction simplifies to $1$. This makes the integral much easier to evaluate.

Step 6: Evaluate the integral

We evaluate the integral of $1$ with respect to $x$, which is $x$. Then, we apply the limits of integration from $0$ to $\frac{\pi}{2}$.

Step 7: Solve for I

Finally, we divide both sides by $2$ to find the value of $I$, which is the original integral.