Find the Integral Using an Appropriate Trigonometric Substitution $$\int \frac { x ^ { 3 } } { \sqrt { x ^ { 2 } + 1 } } d x$$

question 153

Multiple Choice

Find the integral using an appropriate trigonometric substitution. Select the correct answer.
$$\int \frac { x ^ { 3 } } { \sqrt { x ^ { 2 } + 1 } } d x$$

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

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Definitions:

Nociception

The sensory nervous system's process of detecting and transmitting information about stimuli that may cause damage or potential harm to the body.

Perception

The process by which individuals organize and interpret their sensory impressions to give meaning to their environment.

Modulation

The process of varying a signal's properties, such as its amplitude, frequency, or phase, commonly used in telecommunications.

Transduction

The process by which external signals are converted into cellular responses, crucial in various biological systems including sensory perception.