The value of `int_(1)^(e^(37) ) ( sin ( pi log x) )/( x) dx` is `(pi= 3.14)`

Answer

To solve the integral

\[ I = \int_{1}^{e^{37}} \frac{\sin(\pi \log x)}{x} \, dx, \]

we will use a substitution method.

### Step 1: Substitution
Let \( t = \log x \). Then, we have:

\[ x = e^t \quad \text{and} \quad dx = e^t \, dt. \]

### Step 2: Change the limits of integration
When \( x = 1 \):

\[ t = \log(1) = 0, \]

and when \( x = e^{37} \):

\[ t = \log(e^{37}) = 37. \]

Thus, the limits change from \( x = 1 \) to \( x = e^{37} \) into \( t = 0 \) to \( t = 37 \).

### Step 3: Rewrite the integral
Substituting into the integral, we get:

\[ I = \int_{0}^{37} \frac{\sin(\pi t)}{e^t} e^t \, dt = \int_{0}^{37} \sin(\pi t) \, dt. \]

### Step 4: Evaluate the integral
The integral of \( \sin(\pi t) \) is:

\[ \int \sin(\pi t) \, dt = -\frac{1}{\pi} \cos(\pi t) + C. \]

Now we evaluate from \( 0 \) to \( 37 \):

\[ I = \left[-\frac{1}{\pi} \cos(\pi t)\right]_{0}^{37} = -\frac{1}{\pi} \left(\cos(37\pi) - \cos(0)\right). \]

### Step 5: Simplify the expression
Since \( \cos(0) = 1 \) and \( \cos(37\pi) = \cos(\pi) = -1 \) (because \( 37 \) is odd):

\[ I = -\frac{1}{\pi} \left(-1 - 1\right) = -\frac{1}{\pi} \left(-2\right) = \frac{2}{\pi}. \]

### Final Result
Thus, the value of the integral is:

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