QuestionQuestion Evaluate using integration by parts. $$\int x e^{-4 x} d x$$

Studdy Solution
Simplify and solve the remaining integral $\int e^{-4x} \, dx$.
We already found that: $$\int e^{-4x} \, dx = -\frac{1}{4} e^{-4x}$$
Substitute this back into the equation:
$$= -\frac{1}{4} x e^{-4x} + \frac{1}{4} \left(-\frac{1}{4} e^{-4x}\right)$$
$$= -\frac{1}{4} x e^{-4x} - \frac{1}{16} e^{-4x}$$
$$= -\frac{1}{4} e^{-4x} (x + \frac{1}{4}) + C$$
Where $C$ is the constant of integration.
The evaluated integral is:
$$\boxed{-\frac{1}{4} e^{-4x} (x + \frac{1}{4}) + C}$$