When integrating using partial fractions, is there a way to use a calculator to help solve the systems of equations that often emerge?
For example, when given the following problem: $$\int\frac{8+t+6t^2-12t^3}{(3t^2+4)(t^2+7)}dt$$I then got: $$\frac{At+B}{(3t^2+4)}+\frac{Ct+D}{(t^2+7)}$$ Adding and setting the numerators equal got me: $$(At+B)(t^2+7)+(Ct+D)(3t^2+4)=8+t+6t^2-12t^3$$ I then have the following system of equations: $$A+3C=-12$$ $$B+3D=6$$ $$7A+4C=1$$ $$4D+7B=8$$What is the best way to go about solving this systems of equations? And if I have access to a TI-83 and to a computer with internet are there any tools that can help me out?
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$\begingroup$You have substantially two system of equations that are indipendents:$$\begin{cases} A+3C=-12\\7A+4C=1\end{cases}$$ and $$\begin{cases} B+3D=6\\4D+7B=8\end{cases}$$
Both the systems could be solved for example by substitution: $A=-3C-12\implies -21C-84+4C=1\iff C=-5$ and $A=3$. Are you able to solve the second one?
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