Jawab: C

\begin{equation*}
\begin{split}
\int_{-2}^{0} \left(\cos \: (-\pi k x) + \frac {6x^2 - 10x + 7}{k + 2} \right) \: dx & = (k - 2)(k + 7) \\\\
\int_{-2}^{0} \left(\cos \: (-\pi k x) \right) \: dx + \int_{-2}^{0} \left( \frac {6x^2 - 10x + 7}{k + 2} \right) \: dx & = (k - 2)(k + 7) \\\\
\int_{-2}^{0} \left(\cos \: (-\pi k x) \right) \: \frac {d(\cos \: (-\pi k x))}{-\pi k} + \frac {1}{k + 2} \int_{-2}^{0} \left( 6x^2 - 10x + 7 \right) \: dx & = (k - 2)(k + 7) \\\\
\frac {1}{-\pi k} \int_{-2}^{0} \left(\cos \: (-\pi k x) \right) \: d(\cos \: (-\pi k x)) + \frac {1}{k + 2} \int_{-2}^{0} \left( 6x^2 - 10x + 7 \right) \: dx & = (k - 2)(k + 7) \\\\
\frac {1}{-\pi k} \left[ \sin \: (-\pi k x) \right]_{-2}^{0} + \frac {1}{k + 2} \left[2x^3 - 5x^2 + 7x  \right]_{-2}^{0} & = (k - 2)(k + 7) \\\\
\frac {1}{-\pi k} \left[ \sin \: (-\pi k \:.\: 0) - \sin \: (-\pi k \:.\: -2) \right] + \frac {1}{k + 2} \left( \left[2 \:.\: (0)^3 - 5 \:.\: (0)^2 + 7 \:.\: (0)  \right] - \left[2 \:.\: (-2)^3 - 5 \:.\: (-2)^2 + 7 \:.\: (-2)  \right] \right)  & = (k - 2)(k + 7) \\\\
\frac {1}{-\pi k} \left[ \sin 0 - \sin 2 \pi \right] + \frac {1}{k + 2} \left( \left[0  \right] - \left[-16 - 20 - 14  \right] \right)  & = (k - 2)(k + 7) \\\\
\frac {1}{-\pi k} \left[ 0 - 0 \right] + \frac {1}{k + 2} \left( \left[0  \right] - \left[-50  \right] \right)  & = (k - 2)(k + 7) \\\\
\frac {50}{k + 2} & = (k - 2)(k + 7) \\\\
50 & = (k - 2) (k + 2)(k + 7)
\end{split}
\end{equation*}

Menyelesaikan bentuk \((k - 2) (k + 2)(k + 7) = 50\)

Faktor dari 50 adalah 1, 2, 5, 10, 25, 50

Tes nilai k yang sesuai agar menghasilkan faktor-faktor di atas.

Untuk \(k = 3\)

\begin{equation*}
\begin{split}
(k - 2) (k + 2)(k + 7) & = 50 \\\\
(3 - 2) (3 + 2)(3 + 7) & = 50 \\\\
1 \:.\: 5 \:.\: 10 & = 50 \\\\
50 & = 50
\end{split}
\end{equation*}

Nilai k yang sesuai adalah 3

Maka \(\bbox[5px, border: 2px solid magenta] {k + 5 = 8}\)