Express the Integrand as a Sum of Partial Fractions and Evaluate

question 293

Multiple Choice

Express the integrand as a sum of partial fractions and evaluate the integral.
- $$\int \frac { - 2 x ^ { 2 } + 8 x + 8 } { \left( x ^ { 2 } + 4 \right) ( x - 2 ) ^ { 3 } } d x$$

• A) \(\frac { 1 } { 2 } \tan ^ { - 1 } \left( \frac { x } { 2 } \right) + \ln | x - 2 | - \frac { 1 } { ( x - 2 ) ^ { 2 } } + C\)
• B) \(\frac { 1 } { 2 } \tan ^ { - 1 } \left( \frac { x } { 2 } \right) + \frac { 1 } { x - 2 } - \frac { 1 } { ( x - 2 ) ^ { 2 } } + C\)
• C) \(\tan ^ { - 1 } \left( \frac { x } { 2 } \right) - \frac { 2 } { x - 2 } - \frac { 1 } { ( x - 2 ) ^ { 2 } } + C\)
• D) \(\frac { 1 } { 2 } \tan ^ { - 1 } \left( \frac { x } { 2 } \right) - \frac { 1 } { ( x - 2 ) ^ { 2 } } + \frac { 1 } { ( x - 2 ) ^ { 3 } } + C\)

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Definitions:

Somatic Symptom Disorders

A group of disorders in which individuals experience significant physical symptoms that have a psychological origin, often accompanied by excessive thoughts and behaviors related to the symptoms.

Obsessions

Persistent and intrusive thoughts, impulses, or images that cause significant anxiety or distress.

Dissociative Fugue

A disorder characterized by sudden, unplanned travel away from home, combined with an inability to recall one’s past and confusion about personal identity.

Mood Disorders

Psychological disorders characterized by persistent disturbances in mood or emotional state, such as depression or bipolar disorder.