Question

homework9: problem 3
(1 point)
find the antiderivatives for
\\( \frac{d x}{d t}=5 t^{-1}+6 \\).
\\( x=\square+c \\).
hint: in webwork, you write \\( |x| \\) with \abs(x)\.
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Explanation:

Step1: Integrate \(5t^{-1}\)

The integral of \(t^{-1}\) is \(\ln|t|\), so integrating \(5t^{-1}\) gives \(5\ln|t|\).

Step2: Integrate \(6\)

The integral of a constant \(6\) with respect to \(t\) is \(6t\).

Step3: Combine results and add constant

Adding the two integrals together and including the constant of integration \(C\), we get \(x = 5\ln|t| + 6t + C\).

Answer:

\(5\ln|t| + 6t\)