Question

evaluate \\(\int x^2 e^x dx\\)

Explanation:

Identify the integration method

We need to evaluate the indefinite integral:

$$ \int x^2 e^x \, dx $$

Since the integrand is a product of an algebraic function \(x^2\) and an exponential function \(e^x\), we apply Integration by Parts.

Set up the first integration by parts

We use the formula:

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

Let:

$$ u = x^2 \implies du = 2x \, dx $$

$$ dv = e^x \, dx \implies v = e^x $$

Apply the first integration by parts

Substitute \(u\), \(v\), \(du\), and \(dv\) into the formula:

$$ \int x^2 e^x \, dx = x^2 e^x - \int 2x e^x \, dx $$

We factor out the constant:

$$ \int x^2 e^x \, dx = x^2 e^x - 2 \int x e^x \, dx $$

Set up the second integration by parts

We apply integration by parts again to evaluate \(\int x e^x \, dx\).
Let:

$$ u = x \implies du = dx $$

$$ dv = e^x \, dx \implies v = e^x $$

Apply the second integration by parts and simplify

Substitute into the formula for the inner integral:

$$ \int x e^x \, dx = x e^x - \int e^x \, dx = x e^x - e^x $$

Now, substitute this back into our main equation and add the constant of integration \(C\):

$$ \int x^2 e^x \, dx = x^2 e^x - 2(x e^x - e^x) + C $$

$$ \int x^2 e^x \, dx = e^x(x^2 - 2x + 2) + C $$

Answer:

$$ e^x(x^2 - 2x + 2) + C $$