If `I_(n)=int x^(n)e^(x)dx,` then `I_(5)+5I_(4)=`

Options

A. `4x^(4)e^(x)+c`
B. `5x^(4)e^(x)+c`
C. `x^(5)e^(x)+c`
D. `3I_(6)`

Answer

To solve the problem, we need to evaluate the expression \( I_5 + 5I_4 \) where \( I_n = \int x^n e^x \, dx \).

### Step-by-Step Solution:

1. **Define the Integrals:**
\[ I_5 = \int x^5 e^x \, dx \]
\[ I_4 = \int x^4 e^x \, dx \]

2. **Write the Expression:**
We need to find:
\[ I_5 + 5I_4 = \int x^5 e^x \, dx + 5 \int x^4 e^x \, dx \]

3. **Combine the Integrals:**
We can combine the two integrals:
\[ I_5 + 5I_4 = \int x^5 e^x \, dx + \int 5x^4 e^x \, dx = \int (x^5 + 5x^4) e^x \, dx \]

4. **Factor the Expression:**
Notice that \( x^5 + 5x^4 = x^4(x + 5) \):
\[ I_5 + 5I_4 = \int x^4 (x + 5) e^x \, dx \]

5. **Apply Integration by Parts:**
We can use integration by parts. Let:
- \( u = x^4 \) (then \( du = 4x^3 \, dx \))
- \( dv = (x + 5)e^x \, dx \) (then \( v = e^x (x + 5 - 5) = e^x (x + 5) - 5e^x \))

Thus, we can write:
\[ \int u \, dv = uv - \int v \, du \]

6. **Calculate \( uv \):**
\[ uv = x^4 e^x (x + 5) - \int e^x (x + 5) \cdot 4x^3 \, dx \]

7. **Final Expression:**
After performing the integration by parts and simplifying, we find:
\[ I_5 + 5I_4 = e^x (x^5 + 5x^4) - 4I_4 \]

8. **Substituting Back:**
We can substitute back \( I_4 \) to express everything in terms of \( I_4 \) and \( e^x \).

### Final Result:
Thus, we conclude that:
\[ I_5 + 5I_4 = e^x (x^5 + 5x^4) + C \]