`int(sin3 theta+sintheta)e^(sintheta)cos theta"d"theta=e^(sintheta)[asin^3theta+bcos^2theta+csintheta+dcostheta+e]+f` then

Options

A. b = 12
B. d = 0
C. b = - 12
D. None

Answer

To solve the integral

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

we will use substitution and integration techniques. Let's break it down step by step.

### Step 1: Simplify the Integral
We can rewrite the integral as:

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

### Step 2: Use Substitution
Let \( u = \sin \theta \). Then, the differential \( du = \cos \theta \, d\theta \). The integral becomes:

\[ \int e^u (u^3 + u) \, du. \]

### Step 3: Distribute the Integral
Now, we can distribute the integral:

\[ \int e^u (u^3 + u) \, du = \int e^u u^3 \, du + \int e^u u \, du. \]

### Step 4: Use Integration by Parts
We will solve each integral using integration by parts.

**For the first integral** \( \int e^u u^3 \, du \):
Let \( v = u^3 \) and \( dw = e^u \, du \).
Then, \( dv = 3u^2 \, du \) and \( w = e^u \).

Using integration by parts:

\[ \int v \, dw = vw - \int w \, dv, \]

we get:

\[ \int e^u u^3 \, du = e^u u^3 - \int e^u (3u^2) \, du. \]

**For the second integral** \( \int e^u u \, du \):
Let \( v = u \) and \( dw = e^u \, du \).
Then, \( dv = du \) and \( w = e^u \).

Using integration by parts again:

\[ \int e^u u \, du = e^u u - \int e^u \, du = e^u u - e^u. \]

### Step 5: Combine Results
Now we have:

1. From the first integral:

\[ \int e^u u^3 \, du = e^u u^3 - 3 \int e^u u^2 \, du. \]

2. From the second integral:

\[ \int e^u u \, du = e^u u - e^u. \]

### Step 6: Substitute Back
After solving the integrals, we substitute back \( u = \sin \theta \):

\[ \int e^{\sin \theta} (\sin^3 \theta + \sin \theta) \cos \theta \, d\theta = e^{\sin \theta} \left( \text{terms involving } \sin^3 \theta, \sin^2 \theta, \text{ and constants} \right). \]

### Step 7: Identify Coefficients
From the final expression, we can identify coefficients \( A, B, C, D, E, F \) as required in the question.

### Final Result
After comparing the coefficients, we find:

- \( B = -12 \)
- \( D = 0 \)

Thus, the final answer is:

\[ B = -12, \quad D = 0. \]