Evaluate the Integral $$\int \frac { e ^ { t } d t } { e ^ { 2 t } - 6 e ^ { t } + 5 }$$

question 71

Multiple Choice

Evaluate the integral.
- $$\int \frac { e ^ { t } d t } { e ^ { 2 t } - 6 e ^ { t } + 5 }$$

• A) \(\frac { 1 } { 4 } \ln \left| \mathrm { e } ^ { \mathrm { t } } - 5 \right| - \frac { 1 } { 4 } \ln \left| \mathrm { e } ^ { \mathrm { t } } - 1 \right| + \mathrm { C }\)
• B) \(\frac { 1 } { 4 } \ln \left| \mathrm { e } ^ { \mathrm { t } } - 5 \right| + \frac { 1 } { 4 } \ln \left| \mathrm { e } ^ { \mathrm { t } } - 1 \right| + \mathrm { C }\)
• C) \(\frac { 1 } { 4 } \ln | t - 5 | - \frac { 1 } { 4 } \ln | t - 1 | + C\)
• D) \(\frac { 1 } { 4 } \mathrm { e } ^ { \mathrm { t } } \ln \left| \mathrm { e } ^ { \mathrm { t } } - 5 \right| - \frac { 1 } { 4 } \mathrm { e } ^ { \mathrm { t } } \ln \left| \mathrm { e } ^ { \mathrm { t } } - 1 \right| + \mathrm { C }\)

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Definitions:

Alkane Solvents

Solvents composed of alkanes, which are hydrocarbons containing only single bonds between carbon atoms, making them nonpolar and good for dissolving similar substances.

Soluble

The ability of a substance to be dissolved in a solvent, forming a homogeneous mixture at the molecular or ionic level.

Ethers

A class of organic compounds characterized by an oxygen atom connected to two alkyl or aryl groups, widely used as solvents and in the production of pharmaceuticals and chemicals.

Boiling Point

The boiling point is the temperature at which a substance transitions from a liquid to a gas (vaporizes) at a given pressure, distinctive for each substance.