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Mathematics
Problem
Evaluate the definite integral
K
=
∫
0
1
2
x
2
+
3
x
+
1
(
x
+
1
)
(
x
2
+
1
)
d
x
.
K=\int_{0}^{1} \frac{2x^{2}+3x+1}{(x+1)(x^{2}+1)}\,dx.
K
=
∫
0
1
(
x
+
1
)
(
x
2
+
1
)
2
x
2
+
3
x
+
1
d
x
.
(a) Decompose the integrand into partial fractions with denominators
x
+
1
x+1
x
+
1
and
x
2
+
1
x^{2}+1
x
2
+
1
. (b) Hence determine the exact value of
K
K
K
, simplifying your result as far as possible.
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