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

Options

A. 1
B. `-1`
C. 2
D. none

Answer

To solve the integral \( I = \int_{1}^{e^{37}} \frac{\pi \sin(\pi \ln x)}{x} \, dx \), we will use substitution and properties of definite integrals. Here’s a step-by-step solution:

### Step 1: Set up the substitution
Let \( t = \pi \ln x \). Then, differentiating both sides gives:
\[ dt = \frac{\pi}{x} \, dx \quad \Rightarrow \quad dx = \frac{x}{\pi} \, dt = \frac{e^{t/\pi}}{\pi} \, dt \]

### Step 2: Change the limits of integration
When \( x = 1 \):
\[ t = \pi \ln(1) = 0 \]
When \( x = e^{37} \):
\[ t = \pi \ln(e^{37}) = 37\pi \]

### Step 3: Substitute into the integral
Now substitute \( t \) into the integral:
\[ I = \int_{0}^{37\pi} \sin(t) \, dt \]

### Step 4: Integrate
The integral of \( \sin(t) \) is:
\[ \int \sin(t) \, dt = -\cos(t) + C \]
Thus, we evaluate:
\[ I = \left[-\cos(t)\right]_{0}^{37\pi} \]

### Step 5: Evaluate the limits
Now, substituting the limits:
\[ I = -\cos(37\pi) + \cos(0) \]
Since \( \cos(0) = 1 \) and \( \cos(37\pi) = \cos(\pi) = -1 \) (because \( 37 \) is odd):
\[ I = -(-1) + 1 = 1 + 1 = 2 \]

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