Express the Integrand as a Sum of Partial Fractions and Evaluate

question 96

Multiple Choice

Express the integrand as a sum of partial fractions and evaluate the integral.
- $$\int \frac { 75 x ^ { 2 } + 50 x + 3 } { \left( 25 x ^ { 2 } + 1 \right) ^ { 2 } } d x$$

• A) \(\frac { 3 } { 5 } \tan ^ { - 1 } ( 25 x ) + \frac { 1 } { 25 x ^ { 2 } + 1 } + C\)
• B) \(\ln \left| 25 x ^ { 2 } + 1 \right| - \frac { 1 } { 25 x ^ { 2 } + 1 } + C\)
• C) \(\frac { 3 } { 5 } \tan ^ { - 1 } ( 5 x ) - \frac { 1 } { 25 x ^ { 2 } + 1 } + C\)
• D) \(\frac { 3 } { 5 } \tan ^ { - 1 } ( 5 x ) - \frac { 1 } { 25 x ^ { 2 } + 1 } - \frac { 1 } { \left( 25 x ^ { 2 } + 1 \right) ^ { 2 } } + C\)

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Definitions:

Histone Proteins

Proteins that package and order DNA into structural units called nucleosomes, playing a key role in gene regulation and expression.

Linear DNA

DNA molecules that are structured in a straight or nearly straight line, as opposed to being circular.

Nucleosomes

Structures consisting of a segment of DNA wound around a core of histone proteins, playing a crucial role in the packaging of eukaryotic DNA.

DNA Strands

The two long chains of nucleotides that twist around each other to form a double helix structure, which is the physical basis of DNA.