evaluate using integration by parts. (use symbolic notation and fractions where needed. use ( c ) for the…

evaluate using integration by parts. (use symbolic notation and fractions where needed. use ( c ) for the arbitrary constant. absorb into ( c ) as much as possible.) ( int(7 x - 3) e^{-x} d x = )

evaluate using integration by parts. (use symbolic notation and fractions where needed. use ( c ) for the arbitrary constant. absorb into ( c ) as much as possible.) ( int(7 x - 3) e^{-x} d x = )

Answer

Explanation:

Step1: Apply integration by parts formula

The integration by parts formula is (\int u\mathrm{d}v=uv - \int v\mathrm{d}u). Let (u = 7x-3) and (\mathrm{d}v=e^{-x}\mathrm{d}x). Then (\mathrm{d}u = 7\mathrm{d}x) and (v=-e^{-x}). [ \begin{align*} \int(7x - 3)e^{-x}\mathrm{d}x&=(7x - 3)(-e^{-x})-\int(-e^{-x})\times7\mathrm{d}x\ &=-(7x - 3)e^{-x}+7\int e^{-x}\mathrm{d}x \end{align*} ]

Step2: Integrate the remaining integral

We know that (\int e^{-x}\mathrm{d}x=-e^{-x}+C). So, [ \begin{align*} -(7x - 3)e^{-x}+7\int e^{-x}\mathrm{d}x&=-(7x - 3)e^{-x}+7(-e^{-x})+C\ &=-7xe^{-x}+3e^{-x}-7e^{-x}+C\ &=-7xe^{-x}-4e^{-x}+C\ &=-e^{-x}(7x + 4)+C \end{align*} ]

Answer:

(-e^{-x}(7x + 4)+C)