Evaluate the Integral Using the Indicated Trigonometric Substitution $\int \frac { x ^ { 3 } } { \sqrt { x ^ { 2 } + 25 } } d x ; \quad x = 5 \tan \theta$

question 12

Multiple Choice

Evaluate the integral using the indicated trigonometric substitution. $\int \frac { x ^ { 3 } } { \sqrt { x ^ { 2 } + 25 } } d x ; \quad x = 5 \tan \theta$

• A) \(\left( x ^ { 2 } + 25 \right) ^ { 3 / 2 } - 5 \sqrt { x ^ { 2 } + 25 } + C\)
• B) \(\left( x ^ { 2 } + 25 \right) ^ { 3 / 2 } - \sqrt { x ^ { 2 } + 25 } + C\)
• C) \(\frac { 1 } { 3 } \left( x ^ { 2 } + 25 \right) ^ { 3 / 2 } - \sqrt { x ^ { 2 } + 25 } + C\)
• D) \(\frac { 1 } { 3 } \left( x ^ { 2 } + 25 \right) ^ { 3 / 2 } - 25 \sqrt { x ^ { 2 } + 25 } + C\)
• E) \(\frac { 3 } { 2 } ( x + 25 ) ^ { 3 / 2 } - 25 \sqrt { x + 25 } + C\)

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Definitions:

Preoperational Child

Refers to the stage in Piaget’s theory of cognitive development (ages 2-7) where children begin to use language and think symbolically, but their thinking is still intuitive and egocentric.

Conservation

The principle of saving natural resources to prevent depletion and harm to the environment.

Formal Operations

The final stage of cognitive development in Piaget's theory, characterized by logical, systematic thinking and abstract reasoning.

Concrete Operations

A stage in Piaget's theory of cognitive development in which children from about 7 to 12 years old develop logical thinking but still struggle with abstract concepts.