The integral `I=int(e^((e^sinx+sinx)))cos x dx` simpllifies to (where, c is the constant of integration)

Options

A. `e^(sinx)+c`
B. `e^(sinx+cosx)+c`
C. `e^(e^(cosx))+c`
D. `e^(e^(sinx))+c`

Answer

To solve the integral \( I = \int e^{(e^{\sin x} + \sin x)} \cos x \, dx \), we can follow these steps:

### Step 1: Rewrite the Integral
We can rewrite the integral by recognizing that the exponent can be separated:
\[ I = \int e^{e^{\sin x}} e^{\sin x} \cos x \, dx \]

### Step 2: Substitution
Let us use the substitution:
\[ t = e^{\sin x} \]
Then, we differentiate \( t \) with respect to \( x \):
\[ \frac{dt}{dx} = e^{\sin x} \cos x \]
This implies:
\[ dt = e^{\sin x} \cos x \, dx \]
Thus, we can express \( \cos x \, dx \) in terms of \( dt \):
\[ \cos x \, dx = \frac{dt}{e^{\sin x}} \]

### Step 3: Substitute in the Integral
Now substituting \( t \) into the integral:
\[ I = \int e^{t} \, dt \]

### Step 4: Integrate
The integral of \( e^{t} \) is:
\[ I = e^{t} + C \]

### Step 5: Back Substitute
Now we substitute back \( t = e^{\sin x} \):
\[ I = e^{e^{\sin x}} + C \]

### Final Answer
Thus, the integral simplifies to:
\[ I = e^{e^{\sin x}} + C \]

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