The maximum value of `int _(-pi//2) ^(2pi//2) sin x. f (x) dx,` subject to the condition `|f (x) le 5` is M, then `M/10is equal to :

Answer

To solve the problem of finding the maximum value of the integral

\[ M = \int_{-\frac{\pi}{2}}^{\frac{2\pi}{2}} \sin x \cdot f(x) \, dx \]

subject to the condition \(|f(x)| \leq 5\), we can follow these steps:

### Step 1: Understand the Integral and the Condition
The integral is defined from \(-\frac{\pi}{2}\) to \(\pi\) (since \(\frac{2\pi}{2} = \pi\)). The function \(f(x)\) is constrained such that its absolute value does not exceed 5.

### Step 2: Substitute the Maximum Value of \(f(x)\)
To maximize the integral, we can take \(f(x) = 5\) for the entire interval where \(\sin x\) is positive. The sine function is positive between \(0\) and \(\pi\).

### Step 3: Break Down the Integral
The integral can be split into two parts:

\[ M = \int_{-\frac{\pi}{2}}^{0} \sin x \cdot f(x) \, dx + \int_{0}^{\pi} \sin x \cdot f(x) \, dx \]

For the first part, since \(\sin x\) is negative, we can take \(f(x) = -5\) to maximize the negative contribution:

\[ M = \int_{-\frac{\pi}{2}}^{0} \sin x \cdot (-5) \, dx + \int_{0}^{\pi} \sin x \cdot 5 \, dx \]

### Step 4: Evaluate the Integrals
Now we evaluate each integral separately.

1. **First Integral:**
\[ \int_{-\frac{\pi}{2}}^{0} \sin x \cdot (-5) \, dx = -5 \int_{-\frac{\pi}{2}}^{0} \sin x \, dx \]
The integral of \(\sin x\) is \(-\cos x\):
\[ = -5 \left[-\cos x \right]_{-\frac{\pi}{2}}^{0} = -5 \left[-\cos(0) + \cos\left(-\frac{\pi}{2}\right)\right] = -5 \left[-1 + 0\right] = 5 \]

2. **Second Integral:**
\[ \int_{0}^{\pi} \sin x \cdot 5 \, dx = 5 \int_{0}^{\pi} \sin x \, dx \]
Again, using the integral of \(\sin x\):
\[ = 5 \left[-\cos x \right]_{0}^{\pi} = 5 \left[-\cos(\pi) + \cos(0)\right] = 5 \left[1 + 1\right] = 10 \]

### Step 5: Combine the Results
Now we can combine the results of both integrals:

\[ M = 5 + 10 = 15 \]

### Step 6: Find \(M/10\)
Finally, we need to find \(M/10\):

\[ \frac{M}{10} = \frac{15}{10} = 1.5 \]

### Final Answer
Thus, the value of \(M/10\) is:

\[ \frac{M}{10} = 1.5 \]