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Mathematics
Problem
(a) Decompose the expression
x
4
−
x
3
+
2
x
2
−
3
x
+
4
x
3
+
x
2
−
x
−
1
\frac{x^{4}-x^{3}+2x^{2}-3x+4}{x^{3}+x^{2}-x-1}
x
3
+
x
2
−
x
−
1
x
4
−
x
3
+
2
x
2
−
3
x
+
4
into a polynomial part and a proper rational part. (b) Resolve the proper rational part into partial fractions. (c) Using the results of (a) and (b), find
∫
x
4
−
x
3
+
2
x
2
−
3
x
+
4
x
3
+
x
2
−
x
−
1
d
x
.
\int \frac{x^{4}-x^{3}+2x^{2}-3x+4}{x^{3}+x^{2}-x-1}\,dx.
∫
x
3
+
x
2
−
x
−
1
x
4
−
x
3
+
2
x
2
−
3
x
+
4
d
x
.
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