Let $\int x^3 \sin x \, dx = g(x) + C$, where $C$ is the constant of integration. If $8\left(g\left(\frac{\pi}{2}\right) + g'\left(\frac{\pi}{2}\right)\right) = \alpha \pi^3 + \beta \pi^2 + \gamma$, $\alpha, \beta, \gamma \in \mathbb{Z}$, then $\alpha + \beta - \gamma$ equals:

Step 1: Identify g(x) and g'(x)

The problem states that $g(x) + C = \int x^3 \sin x \, dx$. By the Fundamental Theorem of Calculus, the derivative of an integral of a function is the function itself. Therefore, $g'(x)$ is simply the integrand $x^3 \sin x$.

Step 2: Evaluate g'(π/2)

Now we substitute $x = \frac{\pi}{2}$ into the expression for $g'(x)$. We know that $\sin\left(\frac{\pi}{2}\right) = 1$.

Step 3: Integrate g(x) using Integration by Parts

To find $g(x)$, we need to evaluate the integral $\int x^3 \sin x \, dx$. We use integration by parts repeatedly, applying the formula $\int u \, dv = uv - \int v \, du$. We choose $u$ as the polynomial term and $dv$ as the trigonometric term.

Step 4: Evaluate g(π/2)

Now we substitute $x = \frac{\pi}{2}$ into the expression for $g(x)$. We use the values $\cos\left(\frac{\pi}{2}\right) = 0$ and $\sin\left(\frac{\pi}{2}\right) = 1$.

Step 5: Calculate the given expression

Substitute the calculated values of $g\left(\frac{\pi}{2}\right)$ and $g'\left(\frac{\pi}{2}\right)$ into the given expression $8\left(g\left(\frac{\pi}{2}\right) + g'\left(\frac{\pi}{2}\right)\right)$ and simplify.

Step 6: Determine α, β, γ and calculate α + β - γ

By comparing the simplified expression with $\alpha \pi^3 + \beta \pi^2 + \gamma$, we can identify the integer values of $\alpha$, $\beta$, and $\gamma$. Then, we calculate $\alpha + \beta - \gamma$.

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