Evaluate the Integral Using Integration by Parts $\int x e ^ { 3 x } d x$

question 2

Multiple Choice

Evaluate the integral using integration by parts.
- $\int x e ^ { 3 x } d x$

• A) \(\frac { 1 } { 6 }\) \(x ^ { 2 }\) \(e ^ { 3 x }\) + C
• B) \(e ^ { 3 x }\) \(\left( \frac { 1 } { 3 } x - \frac { 1 } { 9 } \right)\) + C
• C) \(e ^ { 3 x }\) \(\left( \frac { 1 } { 3 } x - 1 \right)\) + C
• D) \(\frac { 1 } { 3 }\) \(e ^ { 3 x }\) + C

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Definitions:

Collecting Duct

A series of tubules and ducts in the kidney that process urine, concentrating it before it moves to the bladder.

Cotransport Molecule

A protein that facilitates the simultaneous transport of two molecules across a cell membrane, typically using the gradient of one to transport the other.

Na+

The chemical symbol for sodium ions, which are essential for various physiological functions, including electrical signaling in neurons and muscle cells, as well as fluid balance in the body.

ADH Production

The production of antidiuretic hormone, which helps regulate water balance in the body by controlling the amount of water reabsorbed by the kidneys.