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

Understand the concept of isoquants and the marginal rate of technical substitution (MRTS).

Analyze the implications of varying the input mix on output in the context of isoquants.

Comprehend the relationship between input prices and production choices.

Recognize different types of production functions, including fixed-proportion and variable-proportion systems.

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