Evaluate : `int_(0)^(pi//2) (3^("cos x"))/(3^("sin x") + 3^("cos x")) dx`

Answer

To evaluate the integral

\[ I = \int_0^{\frac{\pi}{2}} \frac{3^{\cos x}}{3^{\sin x} + 3^{\cos x}} \, dx, \]

we will use a property of definite integrals.

### Step 1: Use the substitution property of integrals

We can use the property that states:

\[ \int_a^b f(x) \, dx = \int_a^b f(b - x) \, dx. \]

In our case, we will replace \( x \) with \( \frac{\pi}{2} - x \):

\[ I = \int_0^{\frac{\pi}{2}} \frac{3^{\cos\left(\frac{\pi}{2} - x\right)}}{3^{\sin\left(\frac{\pi}{2} - x\right)} + 3^{\cos\left(\frac{\pi}{2} - x\right)}} \, dx. \]

### Step 2: Simplify the expression

Using the trigonometric identities, we know:

\[ \cos\left(\frac{\pi}{2} - x\right) = \sin x \quad \text{and} \quad \sin\left(\frac{\pi}{2} - x\right) = \cos x. \]

Substituting these into the integral gives:

\[ I = \int_0^{\frac{\pi}{2}} \frac{3^{\sin x}}{3^{\cos x} + 3^{\sin x}} \, dx. \]

### Step 3: Add the two expressions for \( I \)

Now we have two expressions for \( I \):

1. \( I = \int_0^{\frac{\pi}{2}} \frac{3^{\cos x}}{3^{\sin x} + 3^{\cos x}} \, dx \)
2. \( I = \int_0^{\frac{\pi}{2}} \frac{3^{\sin x}}{3^{\cos x} + 3^{\sin x}} \, dx \)

Adding these two equations:

\[ 2I = \int_0^{\frac{\pi}{2}} \left( \frac{3^{\cos x}}{3^{\sin x} + 3^{\cos x}} + \frac{3^{\sin x}}{3^{\cos x} + 3^{\sin x}} \right) \, dx. \]

### Step 4: Combine the fractions

The denominators are the same, so we can combine the numerators:

\[ 2I = \int_0^{\frac{\pi}{2}} \frac{3^{\cos x} + 3^{\sin x}}{3^{\sin x} + 3^{\cos x}} \, dx. \]

This simplifies to:

\[ 2I = \int_0^{\frac{\pi}{2}} 1 \, dx. \]

### Step 5: Evaluate the integral

The integral of 1 from 0 to \( \frac{\pi}{2} \) is simply:

\[ \int_0^{\frac{\pi}{2}} 1 \, dx = \left[ x \right]_0^{\frac{\pi}{2}} = \frac{\pi}{2} - 0 = \frac{\pi}{2}. \]

### Step 6: Solve for \( I \)

Now, substituting back, we have:

\[ 2I = \frac{\pi}{2}. \]

Dividing both sides by 2 gives:

\[ I = \frac{\pi}{4}. \]

Thus, the final result is:

\[ \boxed{\frac{\pi}{4}}. \]