Final answer:
To evaluate the indefinite integral of 2xe^-3x, integration by parts is applied, with u=2x and dv=e^-3x dx resulting in -2/3xe^-3x + 2/3 ∫ e^-3x dx plus a constant of integration C.
Step-by-step explanation:
The student is asking to evaluate the indefinite integral of 2xe-3x. To tackle this integration, we use the integration technique known as integration by parts, which follows from the product rule for differentiation. This technique is expressed as ∫ u dv = uv - ∫ v du, where u and v are functions of x.
For this specific problem: let u = 2x and dv = e-3x dx. After differentiating u and integrating dv, we substitute the resulting functions into the integration by parts formula:
Substituting these into the formula gives the integral as -2/3xe-3x + 2/3 ∫ e-3x dx. The remaining integral is straightforward and the final answer will include the arbitrary constant C, which represents the constant of integration.