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

Answer

To solve the integral \( I = \int_{1}^{e^{37}} \frac{\pi \sin(\pi \log x)}{x} \, dx \), we will use a substitution method.

### Step 1: Substitution
Let \( t = \pi \log x \). Then, we differentiate \( t \) with respect to \( x \):
\[ dt = \frac{\pi}{x} \, dx \quad \Rightarrow \quad dx = \frac{x}{\pi} \, dt \]
Since \( x = e^{t/\pi} \), we can substitute \( dx \) in terms of \( t \):
\[ dx = \frac{e^{t/\pi}}{\pi} \, dt \]

### Step 2: Change the limits of integration
When \( x = 1 \):
\[ t = \pi \log(1) = 0 \]
When \( x = e^{37} \):
\[ t = \pi \log(e^{37}) = 37\pi \]
Thus, the limits change from \( x = 1 \) to \( x = e^{37} \) to \( t = 0 \) to \( t = 37\pi \).

### Step 3: Substitute in the integral
Now substitute \( t \) and \( dx \) into the integral:
\[ I = \int_{0}^{37\pi} \frac{\pi \sin(t)}{e^{t/\pi}} \cdot \frac{e^{t/\pi}}{\pi} \, dt \]
This simplifies to:
\[ I = \int_{0}^{37\pi} \sin(t) \, dt \]

### Step 4: Evaluate the integral
The integral of \( \sin(t) \) is:
\[ \int \sin(t) \, dt = -\cos(t) \]
Thus,
\[ I = \left[-\cos(t)\right]_{0}^{37\pi} = -\cos(37\pi) + \cos(0) \]
Since \( \cos(0) = 1 \) and \( \cos(37\pi) = -1 \) (because \( 37\pi \) is an odd multiple of \( \pi \)):
\[ I = -(-1) + 1 = 1 + 1 = 2 \]

### Final Answer
The value of the integral is:
\[ \boxed{2} \]