`int_(0)^(pi//2) (sin x) / ((sinx + cosx)^(3) ) dx =`

Options

A. `1/4`
B. `(-1)/4`
C. `1/2`
D. `(-1)/2`

Answer

To solve the integral

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

we can use the property of definite integrals. This property states that:

\[ \int_0^a f(x) \, dx = \int_0^a f(a - x) \, dx. \]

### Step 1: Apply the property of definite integrals

Let's apply this property to our integral:

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

Since \(\sin\left(\frac{\pi}{2} - x\right) = \cos x\) and \(\cos\left(\frac{\pi}{2} - x\right) = \sin x\), we can rewrite the integral:

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

### Step 2: Combine the two integrals

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

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

Adding these two equations gives:

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

### Step 3: Simplify the integral

Now we need to compute:

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

### Step 4: Change of variable

Let \(u = \tan x\), then \(du = \sec^2 x \, dx\) and \(\sin x = \frac{u}{\sqrt{1 + u^2}}\), \(\cos x = \frac{1}{\sqrt{1 + u^2}}\). The limits change from \(0\) to \(\infty\) as \(x\) goes from \(0\) to \(\frac{\pi}{2}\).

The integral becomes:

\[ \int_0^{\infty} \frac{1}{\left(\frac{u + 1}{\sqrt{1 + u^2}}\right)^2} \cdot \frac{1}{1 + u^2} \, du. \]

This simplifies to:

\[ \int_0^{\infty} \frac{(1 + u^2)}{(u + 1)^2} \, du. \]

### Step 5: Evaluate the integral

This integral can be evaluated using standard techniques (partial fractions or residue theorem). After evaluating, we find:

\[ \int_0^{\infty} \frac{(1 + u^2)}{(u + 1)^2} \, du = 1. \]

### Step 6: Final result

Thus, we have:

\[ 2I = 1 \implies I = \frac{1}{2}. \]

So the value of the integral is:

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