If `int(3x ^(3)-17)e^(2x)dx={Ax^(3)+Bx^(2)+Cx+D}e^(2x)+k` then

Options

A. `3A+B=0`
B. `B+C=0`
C. `B-D=0`
D. `A+B+C+D=-(71)/(8)`

Answer

To solve the integral \( \int (3x^3 - 17)e^{2x} \, dx \) and express it in the form \( \{Ax^3 + Bx^2 + Cx + D\} e^{2x} + k \), we will use integration by parts multiple times.

### Step 1: Set Up the Integral
We start with the integral:
\[ I = \int (3x^3 - 17)e^{2x} \, dx \]

### Step 2: Split the Integral
We can split the integral into two parts:
\[ I = \int 3x^3 e^{2x} \, dx - 17 \int e^{2x} \, dx \]

### Step 3: Solve the Second Integral
The second integral is straightforward:
\[ \int e^{2x} \, dx = \frac{e^{2x}}{2} + C \]
Thus,
\[ -17 \int e^{2x} \, dx = -\frac{17}{2} e^{2x} \]

### Step 4: Solve the First Integral Using Integration by Parts
For the first integral \( \int 3x^3 e^{2x} \, dx \), we will apply integration by parts. Let:
- \( u = x^3 \) (then \( du = 3x^2 \, dx \))
- \( dv = e^{2x} \, dx \) (then \( v = \frac{e^{2x}}{2} \))

Using the integration by parts formula \( \int u \, dv = uv - \int v \, du \):
\[ \int 3x^3 e^{2x} \, dx = 3 \left( x^3 \cdot \frac{e^{2x}}{2} - \int \frac{e^{2x}}{2} \cdot 3x^2 \, dx \right) \]
This simplifies to:
\[ \frac{3}{2} x^3 e^{2x} - \frac{3}{2} \int x^2 e^{2x} \, dx \]

### Step 5: Solve the New Integral
Now we need to solve \( \int x^2 e^{2x} \, dx \) using integration by parts again:
Let:
- \( u = x^2 \) (then \( du = 2x \, dx \))
- \( dv = e^{2x} \, dx \) (then \( v = \frac{e^{2x}}{2} \))

Applying integration by parts again:
\[ \int x^2 e^{2x} \, dx = x^2 \cdot \frac{e^{2x}}{2} - \int \frac{e^{2x}}{2} \cdot 2x \, dx \]
This simplifies to:
\[ \frac{1}{2} x^2 e^{2x} - \int x e^{2x} \, dx \]

### Step 6: Solve the Last Integral
Now we need to solve \( \int x e^{2x} \, dx \) using integration by parts one more time:
Let:
- \( u = x \) (then \( du = dx \))
- \( dv = e^{2x} \, dx \) (then \( v = \frac{e^{2x}}{2} \))

Applying integration by parts:
\[ \int x e^{2x} \, dx = x \cdot \frac{e^{2x}}{2} - \int \frac{e^{2x}}{2} \, dx \]
This simplifies to:
\[ \frac{1}{2} x e^{2x} - \frac{1}{4} e^{2x} \]

### Step 7: Combine All Parts
Now we can combine all the parts together:
\[ I = \left( \frac{3}{2} x^3 e^{2x} - \frac{3}{2} \left( \frac{1}{2} x^2 e^{2x} - \left( \frac{1}{2} x e^{2x} - \frac{1}{4} e^{2x} \right) \right) - \frac{17}{2} e^{2x} \right) + C \]

### Step 8: Factor Out \( e^{2x} \)
After simplifying, we can factor out \( e^{2x} \):
\[ I = e^{2x} \left( \frac{3}{2} x^3 - \frac{3}{4} x^2 + \frac{3}{4} x - \frac{17}{2} \right) + C \]

### Step 9: Compare Coefficients
Now we can compare coefficients with the form \( \{Ax^3 + Bx^2 + Cx + D\} e^{2x} + k \):
- \( A = \frac{3}{2} \)
- \( B = -\frac{3}{4} \)
- \( C = \frac{3}{4} \)
- \( D = -\frac{17}{2} \)

### Step 10: Solve for Relationships
From the problem, we have the following relationships:
1. \( 3A + B = 0 \)
2. \( B + C = 0 \)
3. \( B - D = 0 \)
4. \( A + B + C + D = -\frac{17}{8} \)