QuestionQuestion Evaluate $\int x e^{5 x} d x$. Choose $u=x$ and $d v=e^{5 x} d x$.

Studdy Solution
Solve the remaining integral $\int e^{5x} \, dx$:
$$\int e^{5x} \, dx = \frac{1}{5} e^{5x}$$
Substitute back into the expression:
$$\frac{x}{5} e^{5x} - \frac{1}{5} \cdot \frac{1}{5} e^{5x}$$
$$= \frac{x}{5} e^{5x} - \frac{1}{25} e^{5x}$$
Combine the terms:
$$= \left(\frac{x}{5} - \frac{1}{25}\right) e^{5x}$$
$$= \frac{5x - 1}{25} e^{5x}$$
Don't forget the constant of integration $C$:
$$= \frac{5x - 1}{25} e^{5x} + C$$
The evaluated integral is:
$$\boxed{\frac{5x - 1}{25} e^{5x} + C}$$