Key Concepts and Formulas

• Integration by substitution: $\int f(g(x))g'(x) \, dx = \int f(u) \, du$, where $u = g(x)$.
• Integration by parts: $\int u \, dv = uv - \int v \, du$.
• The derivative of $x^n$ is $nx^{n-1}$.

Step-by-Step Solution

Step 1: Perform a u-substitution. We are given the integral $\int x^5 e^{-x^2} \, dx$. To simplify this, we perform the substitution $t = x^2$. This implies $dt = 2x \, dx$, or $x \, dx = \frac{1}{2} dt$. Since $x^2 = t$, then $x^4 = t^2$, so $x^5 \, dx = x^4 \cdot x \, dx = t^2 \cdot \frac{1}{2} dt$. Thus, the integral becomes: $\int x^5 e^{-x^2} \, dx = \int t^2 e^{-t} \cdot \frac{1}{2} \, dt = \frac{1}{2} \int t^2 e^{-t} \, dt$

Step 2: Integrate by parts twice. Now we need to evaluate $\int t^2 e^{-t} \, dt$. We will use integration by parts twice. First, let $u = t^2$ and $dv = e^{-t} \, dt$. Then $du = 2t \, dt$ and $v = -e^{-t}$. Using integration by parts, we get: $\int t^2 e^{-t} \, dt = -t^2 e^{-t} - \int (-e^{-t})(2t \, dt) = -t^2 e^{-t} + 2 \int t e^{-t} \, dt$ Now we integrate $\int t e^{-t} \, dt$ by parts. Let $u = t$ and $dv = e^{-t} \, dt$. Then $du = dt$ and $v = -e^{-t}$. So, $\int t e^{-t} \, dt = -te^{-t} - \int (-e^{-t}) \, dt = -te^{-t} + \int e^{-t} \, dt = -te^{-t} - e^{-t} + C_1$ Substituting this back into the first integration by parts, we get: $\int t^2 e^{-t} \, dt = -t^2 e^{-t} + 2(-te^{-t} - e^{-t}) + C_2 = -t^2 e^{-t} - 2te^{-t} - 2e^{-t} + C_2$

Step 3: Substitute back and simplify. Now substitute this back into the original expression: $\frac{1}{2} \int t^2 e^{-t} \, dt = \frac{1}{2} (-t^2 e^{-t} - 2te^{-t} - 2e^{-t}) + C = \left( -\frac{1}{2}t^2 - t - 1 \right) e^{-t} + C$ Finally, substitute $t = x^2$ back into the expression: $\int x^5 e^{-x^2} \, dx = \left( -\frac{1}{2}(x^2)^2 - x^2 - 1 \right) e^{-x^2} + C = \left( -\frac{1}{2}x^4 - x^2 - 1 \right) e^{-x^2} + C$

Step 4: Identify g(x) and evaluate g(-1). We are given that $\int x^5 e^{-x^2} \, dx = g(x) e^{-x^2} + c$. Comparing this with our result, we have: $g(x) = -\frac{1}{2}x^4 - x^2 - 1$ Now we evaluate $g(-1)$: $g(-1) = -\frac{1}{2}(-1)^4 - (-1)^2 - 1 = -\frac{1}{2}(1) - 1 - 1 = -\frac{1}{2} - 2 = -\frac{5}{2}$

Common Mistakes & Tips

• Carefully apply the integration by parts formula, paying attention to signs.
• Don't forget the constant of integration after performing indefinite integration.
• When performing substitution, make sure to substitute back to the original variable at the end.

Summary

We used u-substitution followed by two applications of integration by parts to evaluate the integral $\int x^5 e^{-x^2} \, dx$. After substituting back to the original variable and identifying $g(x)$, we evaluated $g(-1)$, which yielded $-\frac{5}{2}$.

Final Answer

The final answer is \boxed{-5/2}, which corresponds to option (C).