Step 1 : Understanding the Question:
This integration problem involves a product of a polynomial ($x^3$) and an exponential function ($e^x$). The general form of the answer is given, and we are asked to find the specific values of the coefficients $p, q, r$, and $s$ to determine their sum. This type of integral typically requires the repeated application of the "Integration by Parts" technique. The goal is to reduce the degree of the polynomial through successive differentiation until it becomes a constant.
Step 2 : Key Formulas and approach:
The standard Integration by Parts formula is $\int u \, dv = uv - \int v \, du$. For a product of a polynomial and $e^x$, we set $u$ as the polynomial. However, a more efficient approach is the "Tabular Method" (also known as the DI method), which streamlines the process.
The steps are:
1. Differentiate the polynomial term until it reaches zero.
2. Integrate the exponential term an equal number of times.
3. Multiply the terms in pairs along diagonals, alternating the signs starting with positive.
Step 3 : Detailed Explanation:
Let $u = x^3$ and $dv = e^x dx$.
Successive derivatives of $x^3$: $3x^2, 6x, 6, 0$.
Successive integrals of $e^x$: $e^x, e^x, e^x, e^x$.
Combining these with alternating signs: $I = (x^3 \cdot e^x) - (3x^2 \cdot e^x) + (6x \cdot e^x) - (6 \cdot e^x) + C$.
Factor out $e^x$: $I = e^x(x^3 - 3x^2 + 6x - 6) + C$.
Now, we compare this result with the given expression $e^x(px^3 + qx^2 + rx + s) + C$.
From comparison, we find the coefficients: $p = 1, q = -3, r = 6, s = -6$.
The final step is to sum these values: $p + q + r + s = 1 + (-3) + 6 + (-6)$.
Performing the arithmetic: $1 - 3 + 6 - 6 = -2$.
Step 4 : Final Answer:
The sum $p + q + r + s$ is equal to $-2$.