Evaluate ∫ (4/x^3) dx.

To evaluate the integral of (\frac{4}{x^3}) with respect to (x), we can rewrite the integrand in a more manageable form. The expression (\frac{4}{x^3}) can be expressed as (4x^{-3}).

Now, we can apply the power rule of integration, which states that (\int x^n , dx = \frac{x^{n+1}}{n+1} + C) for (n \neq -1).

Here, (n = -3), so we apply the power rule:

[

\int 4x^{-3} , dx = 4 \cdot \frac{x^{-3 + 1}}{-3 + 1} + C = 4 \cdot \frac{x^{-2}}{-2} + C

]

This simplifies to:

[

4 \cdot \left(-\frac{1}{2} x^{-2}\right) + C = -2x^{-2} + C

]

Reverting (x^{-2}) back to its original form ( \frac{1}{x^2} ), we find: