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

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Definitions:

Venous Hum

A continuous, low-pitched sound heard over the major veins, particularly in the neck, due to turbulent blood flow.

Murmur

Abnormal heart sounds during a heartbeat cycle, which can indicate various heart conditions.

Carotid Artery

A major blood vessel in the neck that supplies blood to the brain, neck, and face.

Bruits

Sounds heard over the arteries or veins during auscultation, indicative of abnormal blood flow, often due to narrow or blocked vessels.