Question

evaluate the indefinite integral.\\(int \frac{2}{(t + 3)^4} dt\\)\\(\boxed{} + c\\)

Explanation:

Step1: Set substitution variable

Let $u = t + 3$, so $du = dt$.

Step2: Rewrite integral in terms of u

Integral becomes $2\int u^{-4} du$.

Step3: Apply power rule for integration

$\int u^n du = \frac{u^{n+1}}{n+1} + C$ (n≠-1). Here n=-4:
$2 \cdot \frac{u^{-3}}{-3} + C = -\frac{2}{3}u^{-3} + C$.

Step4: Substitute back u = t+3

$-\frac{2}{3(t+3)^3} + C$.

Answer:

$-\frac{2}{3(t+3)^3}$