The value of `inte^(x)(x^(5)+5x^(4)+1)dx` is

Options

A. `e^(x).x^(5)+C`
B. `e^(x).x^(5)+e^(x)+C`
C. `e^(x+1).e^(5)+C`
D. `5x^(4).e^(x)+C`

Explanation

To solve the integral \( \int e^x (x^5 + 5x^4 + 1) \, dx \), we can use integration by parts. Let's go through the steps:

### Step 1: Set up the integral
Let:
\[ I = \int e^x (x^5 + 5x^4 + 1) \, dx \]

### Step 2: Break down the integral
We can break down the integral into three separate integrals:
\[ I = \int e^x x^5 \, dx + 5 \int e^x x^4 \, dx + \int e^x \, dx \]

### Step 3: Solve the first integral using integration by parts
For \( \int e^x x^5 \, dx \), we apply integration by parts:
Let \( u = x^5 \) and \( dv = e^x \, dx \).
Then, \( du = 5x^4 \, dx \) and \( v = e^x \).

Using the integration by parts formula \( \int u \, dv = uv - \int v \, du \):
\[ \int e^x x^5 \, dx = e^x x^5 - \int e^x (5x^4) \, dx \]

### Step 4: Solve the second integral
Now we need to evaluate \( \int e^x (5x^4) \, dx \):
Using integration by parts again:
Let \( u = x^4 \) and \( dv = e^x \, dx \).
Then, \( du = 4x^3 \, dx \) and \( v = e^x \).

So,
\[ \int e^x (5x^4) \, dx = 5 \left( e^x x^4 - \int e^x (4x^3) \, dx \right) \]

### Step 5: Continue this process
We continue this process for \( \int e^x (4x^3) \, dx \), \( \int e^x (3x^2) \, dx \), \( \int e^x (2x) \, dx \), and \( \int e^x \, dx \) until we reach the base case \( \int e^x \, dx \).

### Step 6: Combine all results
After performing integration by parts repeatedly, we will find that:
\[ I = e^x \left( x^5 + 5x^4 + 20x^3 + 60x^2 + 120x + 120 \right) + C \]

### Final Answer
Thus, the value of the integral is:
\[ \int e^x (x^5 + 5x^4 + 1) \, dx = e^x \left( x^5 + 5x^4 + 20x^3 + 60x^2 + 120x + 120 \right) + C \]