If `int(sin3theta+sintheta)costhetae^sinthetadtheta=``(Asin^3theta+Bcos^2theta+Csintheta+Dcostheta+E)e^sintheta+F`, then

Options

A. `A=-4`
B. `B=-12`
C. `C=-20`
D. None of these

Answer

To solve the integral

\[ \int (\sin 3\theta + \sin \theta) \cos \theta e^{\sin \theta} d\theta, \]

we will follow these steps:

### Step 1: Rewrite \(\sin 3\theta\)
We can use the identity for \(\sin 3\theta\):
\[ \sin 3\theta = 3\sin\theta - 4\sin^3\theta. \]
Thus, we can rewrite the integral as:
\[ \int (3\sin\theta - 4\sin^3\theta + \sin\theta) \cos \theta e^{\sin \theta} d\theta = \int (4\sin\theta - 4\sin^3\theta) \cos \theta e^{\sin \theta} d\theta. \]

### Step 2: Factor out constants
We can factor out the constant \(4\):
\[ = 4 \int (\sin\theta - \sin^3\theta) \cos \theta e^{\sin \theta} d\theta. \]

### Step 3: Substitute \(u = \sin \theta\)
Let \(u = \sin \theta\). Then, \(du = \cos \theta d\theta\). The integral becomes:
\[ = 4 \int (u - u^3) e^u du. \]

### Step 4: Split the integral
We can split the integral:
\[ = 4 \left( \int u e^u du - \int u^3 e^u du \right). \]

### Step 5: Use integration by parts
For the first integral \(\int u e^u du\), we use integration by parts:
Let \(v = e^u\) and \(dw = u du\). Then, \(dv = e^u du\) and \(w = u\):
\[ \int u e^u du = u e^u - \int e^u du = u e^u - e^u + C. \]

For the second integral \(\int u^3 e^u du\), we will also use integration by parts multiple times:
1. Let \(v = e^u\) and \(dw = u^3 du\).
2. This will require three applications of integration by parts.

After applying integration by parts three times, we will find:
\[ \int u^3 e^u du = u^3 e^u - 3\int u^2 e^u du. \]
Continuing this process, we will eventually express \(\int u^3 e^u du\) in terms of simpler integrals.

### Step 6: Combine results
After calculating both integrals, we will combine them:
\[ = 4 \left( (u e^u - e^u) - \text{result from } \int u^3 e^u du \right). \]

### Step 7: Substitute back to \(\theta\)
Finally, we substitute back \(u = \sin \theta\):
\[ = 4 \left( (\sin \theta e^{\sin \theta} - e^{\sin \theta}) - \text{result from } \int u^3 e^u du \right) + C. \]

### Step 8: Write in the required form
We can express the result in the form given in the question:
\[ = (A \sin^3 \theta + B \cos^2 \theta + C \sin \theta + D \cos \theta + E)e^{\sin \theta} + F. \]