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Unfortunately, this means that the site will be down during this time. I apologize for any inconvenience this might cause.
Paul
February 18, 2026
Section 7.1 : Integration by Parts
9. Evaluate \( \displaystyle \int{{\left( {4{x^3} - 9{x^2} + 7x + 3} \right){{\bf{e}}^{ - x}}\,dx}}\) .
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Okay, with this problem doing the “standard” method of integration by parts (i.e. picking \(u\) and \(dv\) and using the formula) would take quite a bit of time. So, this looks like a good problem to use the table that we saw in the notes to shorten the process up.
Here is the table for this problem.
\[\begin{array}{rrr} 4{{x}^{3}}-9{{x}^{2}}+7x+3 & {{\mathbf{e}}^{-x}} & + \\ 12{{x}^{2}}-18x+7 & -{{\mathbf{e}}^{-x}} & - \\ 24x-18 & {{\mathbf{e}}^{-x}} & + \\ 24 & -{{\mathbf{e}}^{-x}} & - \\ 0 & {{\mathbf{e}}^{-x}} & + \\ \end{array}\] Show Step 2Here’s the integral for this problem,
\[\begin{align*}\int{{\left( {4{x^3} - 9{x^2} + 7x + 3} \right){{\bf{e}}^{ - x}}\,dx}} & = \left( {4{x^3} - 9{x^2} + 7x + 3} \right)\left( { - {{\bf{e}}^{ - x}}} \right) - \left( {12{x^2} - 18x + 7} \right)\left( {{{\bf{e}}^{ - x}}} \right)\\ & \hspace{0.5in} + \left( {24x - 18} \right)\left( { - {{\bf{e}}^{ - x}}} \right) - \left( {24} \right)\left( {{{\bf{e}}^{ - x}}} \right) + c\\ & = - {{\bf{e}}^{ - x}}\left( {4{x^3} - 9{x^2} + 7x + 3} \right) - {{\bf{e}}^{ - x}}\left( {12{x^2} - 18x + 7} \right)\\ & \hspace{0.5in} - {{\bf{e}}^{ - x}}\left( {24x - 18} \right) - 24{{\bf{e}}^{ - x}} + c\\ & = \require{bbox} \bbox[2pt,border:1px solid black]{{ - {{\bf{e}}^{ - x}}\left( {4{x^3} + 3{x^2} + 13x + 16} \right)+c}}\end{align*}\]