If \( g(x)=\frac{f(x)}{(x-a)(x-b)(x-c)} \), where \( f(x) \) is a polynomial of degree \( 3 \), ...
If \( g(x)=\frac{f(x)}{(x-a)(x-b)(x-c)} \), where \( f(x) \) is a polynomial of degree \( 3 \), then
(a) \( \int g(x) d x=\left|\begin{array}{lll}1 & a & f(a) \log \mid x-a \\ 1 & b & f(b) \log \mid x-b \\ 1 & c & f(c) \log \mid x-c\end{array}\right| \div\left|\begin{array}{lll}1 & a & a^{2} \\ 1 & b & b^{2} \\ 1 & c & c^{2}\end{array}\right|+k \)
(b) \( \frac{d g(x)}{d x}=\left|\begin{array}{lll}1 & a & f(a)(x-a)^{-2} \\ 1 & b & f(b)(x-b)^{-2} \\ 1 & c & f(c)(x-c)^{-2}\end{array}\right| \div\left|\begin{array}{lll}a^{2} & a & 1 \\ b^{2} & b & 1 \\ c^{2} & c & 1\end{array}\right| \)
(c) \( \frac{d g(x)}{d x}=\left|\begin{array}{lll}1 & a & f(a)(x-a)^{-2} \\ 1 & b & f(b)(x-b)^{-2} \\ 1 & c & f(c)(x-c)^{-2}\end{array}\right| \div\left|\begin{array}{ccc}1 & a & a^{2} \\ 1 & b & b^{2} \\ 1 & c & c^{2}\end{array}\right| \)
(d) \( \int g(x) d x=\left|\begin{array}{lll}1 & a & f(a) \log \mid x-a \\ 1 & b & f(b) \log \mid x-b\end{array}\right| \div\left|\begin{array}{lll}a^{2} & a & 1 \\ b^{2} & b & 1 \\ 1 & c & f(c) \log \mid x-c\end{array}\right|+k \)
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