Express the Integrand as a Sum of Partial Fractions and Evaluate

question 209

Multiple Choice

Express the integrand as a sum of partial fractions and evaluate the integral.
- $$\int \frac { 4 x ^ { 3 } + 20 x ^ { 2 } + 32 x + 13 } { x ^ { 2 } \left( x ^ { 2 } + 6 x + 13 \right) } d x$$

• A) \(7 \ln | x | + \ln \left| x ^ { 2 } + 6 x + 13 \right| + \frac { 1 } { 2 } \tan ^ { - 1 } \left( \frac { x + 3 } { 2 } \right) + C\)
• B) \(2 \ln | x | - \frac { 1 } { x } + \ln \left| x ^ { 2 } + 6 x + 13 \right| + \frac { 1 } { 2 } \tan ^ { - 1 } \left( \frac { x + 3 } { 2 } \right) + C\)
• C) \(2 \ln \left| x - \frac { 1 } { x } \right| + \ln \left| x ^ { 2 } + 6 x + 13 \right| + \frac { 1 } { 2 } \tan ^ { - 1 } \left( \frac { x + 3 } { 2 } \right) + C\)
• D) \(5 \ln | x | + \frac { 1 } { x } - \ln \left| x ^ { 2 } + 6 x + 13 \right| + \sin ^ { - 1 } \left( \frac { x + 3 } { 2 } \right) + C\)

---

Definitions:

Infidelity

The act of being unfaithful in a romantic relationship, typically referring to sexual relationships outside of one's committed partnership.

Extramarital Sex

Sexual activities conducted with someone other than one's spouse, often viewed through the lens of infidelity or unfaithfulness.

Emotional Infidelity

A breach of trust involving forming a deep emotional connection with someone outside the relationship, without necessarily involving physical intimacy.

Adjusting Entry

A journal entry made at the end of an accounting period to allocate income and expenditure to the appropriate years.