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

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Definitions:

Microsaccades

Small, involuntary eye movements that occur during focus, helping to refresh the image on the retina and prevent visual fading.

Sensory Adaptation

The process by which sensory receptors become less sensitive to constant stimuli over time.

Microsaccades

Small, involuntary eye movements that occur when the gaze is fixed, helping in refreshing the image on the retina to prevent visual fading.

Optic Chiasm

The optic chiasm is a crucial part of the brain where the optic nerves partially cross, allowing visual information from both eyes to be processed by both hemispheres of the brain.