Match each function to its antiderivative, where $t3$ and $C$ is a constant.
\begin{tabular}{c|c}
Antiderivative & Function \
\hline$H(t)=\ln \left(12 t^{2}-12 ight)+C$ & $h(t)=\frac{3 t^{2}-3}{t^{3}-3 t}$ \
$H(t)=\ln \left(t^{4}-6 t^{2} ight)+C$ & $h(t)=\frac{4 t^{2}-12}{t^{3}-6 t}$ \
$H(t)=\ln \left(4 t^{3}-12 t ight)+C$ & $h(t)=\frac{2 t}{t^{2}-1}$
\end{tabular}

Question

Match each function to its antiderivative, where $t>3$ and $C$ is a constant.
\begin{tabular}{c|c} 
Antiderivative & Function \
\hline$H(t)=\ln \left(12 t^{2}-12ight)+C$ & $h(t)=\frac{3 t^{2}-3}{t^{3}-3 t}$ \
$H(t)=\ln \left(t^{4}-6 t^{2}ight)+C$ & $h(t)=\frac{4 t^{2}-12}{t^{3}-6 t}$ \
$H(t)=\ln \left(4 t^{3}-12 tight)+C$ & $h(t)=\frac{2 t}{t^{2}-1}$
\end{tabular}

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Identify the given functions and their potential antiderivatives from the table. ‖ ‖ step 2 ⋮ For the function $h(t) = \frac{3 t^{2} - 3}{t^{3} - 3 t}$, simplify it to see if it matches any antiderivative: $$ h(t) = \frac{3(t^{2} - 1)}{t(t^{2} - 3)} = \frac{3}{t} \cdot \frac{t^{2} - 1}{t^{2} - 3} $$ This does not simplify directly to a known form. ‖ ‖ step 3 ⋮ For the function $h(t) = \frac{4 t^{2} - 12}{t^{3} - 6 t}$, simplify it: $$ h(t) = \frac{4(t^{2} - 3)}{t(t^{2} - 6)} = \frac{4}{t} \cdot \frac{t^{2} - 3}{t^{2} - 6} $$ This also does not simplify directly to a known form. ‖ ‖ step 4 ⋮ For the function $h(t) = \frac{2 t}{t^{2} - 1}$, simplify it: $$ h(t) = \frac{2 t}{(t - 1)(t + 1)} $$ This matches the derivative of $H(t) = \ln(4 t^{3} - 12 t) + C$ because: $$ \frac{d}{dt} \ln(4 t^{3} - 12 t) = \frac{12 t^{2} - 12}{4 t^{3} - 12 t} = \frac{3 t^{2} - 3}{t^{3} - 3 t} $$ Thus, $H(t) = \ln(4 t^{3} - 12 t) + C$ is the antiderivative of $h(t) = \frac{2 t}{t^{2} - 1}$. ‖ ‖ step 5 ⋮ Verify the remaining functions and antiderivatives: - $H(t) = \ln(12 t^{2} - 12) + C$ matches $h(t) = \frac{3 t^{2} - 3}{t^{3} - 3 t}$ - $H(t) = \ln(t^{4} - 6 t^{2}) + C$ matches $h(t) = \frac{4 t^{2} - 12}{t^{3} - 6 t}$ ‖ ∻Answer∻ ⚹ The correct matches are: - $H(t) = \ln(12 t^{2} - 12) + C$ with $h(t) = \frac{3 t^{2} - 3}{t^{3} - 3 t}$ - $H(t) = \ln(t^{4} - 6 t^{2}) + C$ with $h(t) = \frac{4 t^{2} - 12}{t^{3} - 6 t}$ - $H(t) = \ln(4 t^{3} - 12 t) + C$ with $h(t) = \frac{2 t}{t^{2} - 1}$ ⚹ ∻Key Concept∻ ⚹Matching functions to their antiderivatives⚹ ∻Explanation∻ ⚹To match a function to its antiderivative, simplify the function and compare it to the derivative of the given antiderivative.⚹◊What is the definition of an antiderivative in calculus?⍭ Generate me a similar question◊

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