Question 11

Evaluate the indefinite integral.
\[
\int \frac{4}{(t+4)^{2}} d t=\square+c
\]

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Answer

$\boxed{-4/(t + 4) + C}$ is the solution to the integral.

Steps

Step 1 ：Let's set $u = t + 4$. Then, $du = dt$.

Step 2 ：Substitute these into the integral: $\int \frac{4}{u^{2}} du$.

Step 3 ：This integral can be rewritten as: $4 \int u^{-2} du$.

Step 4 ：Now we can apply the power rule for integration: $4 \cdot \frac{1}{-1} u^{-1} + C$.

Step 5 ：Simplify this to: $-4u^{-1} + C$.

Step 6 ：Finally, substitute $t + 4$ back in for $u$ to get the answer in terms of $t$: $-4/(t + 4) + C$.

Step 7 ：$\boxed{-4/(t + 4) + C}$ is the solution to the integral.