Before doing the integral we need to break up the quotient and do some simplification.

\[\int{{{t^3} - \frac{{{{\bf{e}}^{ - t}} - 4}}{{{{\bf{e}}^{ - t}}}}\,dt}} = \int{{{t^3} - \frac{{{{\bf{e}}^{ - t}}}}{{{{\bf{e}}^{ - t}}}} + \frac{4}{{{{\bf{e}}^{ - t}}}}\,dt}} = \int{{{t^3} - 1 + 4{{\bf{e}}^t}\,dt}}\]

Make sure that you correctly distribute the minus sign when breaking up the second term and don’t forget to move the exponential in the denominator of the third term (after splitting up the integrand) to the numerator and changing the sign on the \(t\) to a “+” in the process.

Show Step 2

At this point there really isn’t too much to do other than to evaluate the integral.

\[\int{{{t^3} - \frac{{{{\bf{e}}^{ - t}} - 4}}{{{{\bf{e}}^{ - t}}}}\,dt}} = \int{{{t^3} - 1 + 4{{\bf{e}}^t}\,dt}} = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{1}{4}{t^4} - t + 4{{\bf{e}}^t} + c}}\]