Evaluate the Integral by Making a Substitution (Possibly Trigonometric) and Then

question 280

Multiple Choice

Evaluate the integral by making a substitution (possibly trigonometric) and then applying a reduction formula.
- $$\int e ^ { t } \sec ^ { 3 } \left( e ^ { t } - 2 \right) d t$$

• A) \(- \frac { 1 } { 2 } \left[ \sec \left( e ^ { t } - 2 \right) \tan \left( e ^ { t } - 2 \right) - \ln \left| \sec \left( e ^ { t } - 2 \right) + \tan \left( e ^ { t } - 2 \right) \right| \right] + C\)
• B) \(\frac { 1 } { 2 } \left[ \sec \left( e ^ { t } - 2 \right) \tan \left( e ^ { t } - 2 \right) - \ln \left| \sec \left( e ^ { t } - 2 \right) + \tan \left( e ^ { t } - 2 \right) \right| \right] + C\)
• C) \(- \frac { 1 } { 2 } \left[ \sec \left( e ^ { t } - 2 \right) \tan \left( e ^ { t } - 2 \right) + \ln \left| \sec \left( e ^ { t } - 2 \right) + \tan \left( e ^ { t } - 2 \right) \right| \right] + C\)
• D) \(\frac { 1 } { 2 } \left[ \sec \left( e ^ { t } - 2 \right) \tan \left( e ^ { t } - 2 \right) + \ln \left| \sec \left( e ^ { t } - 2 \right) + \tan \left( e ^ { t } - 2 \right) \right| \right] + C\)