Express the Integrand as a Sum of Partial Fractions and Evaluate

question 222

Multiple Choice

Express the integrand as a sum of partial fractions and evaluate the integral.
- $$\int \frac { 2 x ^ { 2 } + x + 1 } { \left( x ^ { 2 } + 3 \right) ( x - 5 ) } d x$$

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

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Definitions:

Predetermined Overhead Rate

A calculated rate used to allocate manufacturing overhead costs to individual units of production, based on estimated costs and activity levels.

Direct Labor-hours

The total hours of labor directly involved in producing goods, used as a measure for costing and operational efficiency.

Manufacturing Overhead

All manufacturing costs that are not directly related to the production of goods or services, such as factory rent, utilities, and equipment maintenance.

Predetermined Overhead Rate

A rate calculated before a cost period begins, used to allocate overhead costs to products based on a chosen activity base.