Evaluate the Integral $$\int \frac { ( x + 7 ) ^ { 2 } \tan ^ { - 1 } x + ( 4 x - 22 ) ( x + 7 ) } { \left( x ^ { 2 } + 1 \right) ( x + 7 ) ^ { 2 } } d x$$

question 329

Multiple Choice

Evaluate the integral.
- $$\int \frac { ( x + 7 ) ^ { 2 } \tan ^ { - 1 } x + ( 4 x - 22 ) ( x + 7 ) } { \left( x ^ { 2 } + 1 \right) ( x + 7 ) ^ { 2 } } d x$$

• A) \(\frac { \left( \tan ^ { - 1 } x \right) ^ { 2 } } { 2 } - \ln | x + 7 | + \frac { 1 } { 2 } \ln \left( x ^ { 2 } + 1 \right) + C\)
• B) \(\frac { \left( \tan ^ { - 1 } x \right) ^ { 2 } } { 2 } - \ln | x + 7 | - 5 \tan ^ { - 1 } x + \ln \left( x ^ { 2 } + 1 \right) + C\)
• C) \(\frac { \left( \tan ^ { - 1 } x \right) ^ { 2 } } { 2 } - 3 \tan ^ { - 1 } x + \frac { 1 } { 2 } \ln \left( x ^ { 2 } + 1 \right) + C\)
• D) \(\frac { \left( \tan ^ { - 1 } x \right) ^ { 2 } } { 2 } - \ln | x + 7 | - 3 \tan ^ { - 1 } x + \frac { 1 } { 2 } \ln \left( x ^ { 2 } + 1 \right) + C\)

---

Definitions:

Agoraphobia

A fear or anxiety disorder characterized by intense fear of situations where escape might be difficult or where help wouldn't be available if things go wrong, often involving fear of open or enclosed spaces, crowds, or leaving home alone.

Affective Disorder

A category of mental health disorders that primarily affect mood and emotions, such as depression and bipolar disorder.

Somatoform Disorder

A type of mental disorder characterized by the presence of physical symptoms that cannot be fully explained by a general medical condition, substance use, or another mental disorder.

Paranoia

A psychiatric disorder marked by unfounded suspicions of being targeted, excessive jealousy without cause, or an overinflated perspective of one's own significance.