Let `d/(dx)F(x)=((e^(sinx))/x),x > 0.` If `int_1^4 3/x e^sin x^3dx=F(k)-F(1),` then one of the possible values of `k ,` is: 15 (b) 16 (c) 63 (d) 64

Options

A. 16
B. 63
C. 64
D. 15

Answer

To solve the given problem step by step, we will follow the instructions provided in the video transcript and derive the solution systematically.

### Step 1: Understand the given information
We are given:
\[ \frac{d}{dx} F(x) = \frac{e^{\sin x}}{x}, \quad x > 0 \]
We also have the integral:
\[ \int_1^4 \frac{3}{x} e^{\sin x^3} \, dx = F(k) - F(1) \]

### Step 2: Manipulate the integral
To solve the integral, we will multiply and divide by \(x^2\):
\[ \int_1^4 \frac{3}{x} e^{\sin x^3} \, dx = \int_1^4 \frac{3x^2}{x^3} e^{\sin x^3} \, dx \]

### Step 3: Change of variables
Let \(t = x^3\). Then, we differentiate:
\[ dt = 3x^2 \, dx \quad \Rightarrow \quad dx = \frac{dt}{3x^2} \]
When \(x = 1\), \(t = 1^3 = 1\).
When \(x = 4\), \(t = 4^3 = 64\).

Substituting these into the integral:
\[ \int_1^4 \frac{3}{x} e^{\sin x^3} \, dx = \int_1^{64} \frac{e^{\sin t}}{t} \, dt \]

### Step 4: Relate the integral to \(F(k) - F(1)\)
From the original equation, we have:
\[ \int_1^{64} \frac{e^{\sin t}}{t} \, dt = F(k) - F(1) \]
Since \(\frac{d}{dx} F(x) = \frac{e^{\sin x}}{x}\), we can express the integral as:
\[ F(64) - F(1) = F(k) - F(1) \]

### Step 5: Equate and solve for \(k\)
Since \(F(1)\) cancels out from both sides, we have:
\[ F(64) = F(k) \]
This implies that \(k = 64\).

### Final Answer
Thus, one of the possible values of \(k\) is:
\[ \boxed{64} \]