Question

determine
(a) (int 4 e^{-3 x} d x)

Explanation:

Step1: Use Constant Multiple Rule

The integral of a constant multiple of a function is the constant multiple of the integral of the function. So, $\int 4e^{-3x} dx = 4\int e^{-3x} dx$.

Step2: Use Substitution for Exponential Integral

Let $u = -3x$, then $du = -3dx$ or $dx = \frac{-1}{3}du$. Substituting into the integral, we get $4\int e^{u} \cdot \frac{-1}{3}du$.

Step3: Simplify and Integrate

Simplify the integral: $4 \cdot \frac{-1}{3}\int e^{u} du = -\frac{4}{3}e^{u} + C$.

Step4: Substitute Back

Substitute back $u = -3x$: $-\frac{4}{3}e^{-3x} + C$.

Answer:

$\boxed{-\dfrac{4}{3}e^{-3x} + C}$