Question

evaluate the indefinite integral.\\(\int \frac{2}{(t + 12)^8} dt\\)\\(\boxed{}\\) + c\
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Explanation:

Step1: Use substitution method

Let $u = t + 12$, then $du = dt$.

Step2: Rewrite integral in terms of u

$$\int \frac{2}{u^8} du = 2\int u^{-8} du$$

Step3: Apply power rule for integration

The power rule is $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ for $n
eq -1$.
$$2 \cdot \frac{u^{-8+1}}{-8+1} + C = 2 \cdot \frac{u^{-7}}{-7} + C$$

Step4: Substitute back $u = t+12$

$$2 \cdot \frac{(t+12)^{-7}}{-7} + C = -\frac{2}{7(t+12)^7} + C$$

Answer:

$-\frac{2}{7(t+12)^7}$