We are tasked with evaluating the integral:

$\int 9x^2 e^{-2x} \, dx$

Step 1: Use Integration by Parts

The formula for integration by parts is:

$\int u \, dv = uv - \int v \, du$

Choose $u$ and $dv$:

Let:

• $u = 9x^2$, so $du = 18x \, dx$
• $dv = e^{-2x} \, dx$, so $v = \int e^{-2x} \, dx = -\frac{1}{2}e^{-2x}$

Step 2: Apply the formula

Substituting into the integration by parts formula:

$\int 9x^2 e^{-2x} \, dx = 9x^2 \cdot \left(-\frac{1}{2}e^{-2x}\right) - \int \left(-\frac{1}{2}e^{-2x}\right) \cdot 18x \, dx$

Simplify:

$\int 9x^2 e^{-2x} \, dx = -\frac{9x^2}{2} e^{-2x} + 9 \int x e^{-2x} \, dx$

Step 3: Repeat Integration by Parts for $\int x e^{-2x}$

Let:

• $u = x$, so $du = dx$
• $dv = e^{-2x} \, dx$, so $v = -\frac{1}{2}e^{-2x}$

Substituting into the formula:

$\int x e^{-2x} \, dx = x \cdot \left(-\frac{1}{2}e^{-2x}\right) - \int \left(-\frac{1}{2}e^{-2x}\right) \cdot 1 \, dx$

Simplify:

$\int x e^{-2x} \, dx = -\frac{x}{2} e^{-2x} + \frac{1}{2} \int e^{-2x} \, dx$

The integral $\int e^{-2x} \, dx$ evaluates to $-\frac{1}{2}e^{-2x}$. Substituting:

$\int x e^{-2x} \, dx = -\frac{x}{2} e^{-2x} - \frac{1}{4} e^{-2x}$

Step 4: Substitute back

Returning to the original integral:

$\int 9x^2 e^{-2x} \, dx = -\frac{9x^2}{2} e^{-2x} + 9 \left(-\frac{x}{2} e^{-2x} - \frac{1}{4} e^{-2x}\right)$

Simplify:

$\int 9x^2 e^{-2x} \, dx = -\frac{9x^2}{2} e^{-2x} - \frac{9x}{2} e^{-2x} - \frac{9}{4} e^{-2x}$

Final Answer:

$\int 9x^2 e^{-2x} \, dx = -\frac{9}{2} x^2 e^{-2x} - \frac{9}{2} x e^{-2x} - \frac{9}{4} e^{-2x} + C$

where $C$ is the constant of integration.

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Would you like further details or examples? Here are 5 related questions to consider:

1. What is the general method for solving integrals involving $x^n e^{ax}$?
2. How does integration by parts apply to polynomial and exponential combinations?
3. Can you derive the formula for repeated integration by parts?
4. How do exponential decay terms affect the convergence of integrals over an infinite domain?
5. What are the conditions under which integration by parts is most efficient?

Tip: Always simplify as much as possible before applying a second integration by parts step!