Answer
$48,620x^{18}$
Work Step by Step
$(a+b)^{n}=\left(\begin{array}{l}
n\\
0
\end{array}\right)a^{n}+\left(\begin{array}{l}
n\\
1
\end{array}\right)a^{n-1}b+\left(\begin{array}{l}
n\\
2
\end{array}\right)a^{n-2}b^{2}+\cdots+\left(\begin{array}{l}
n\\
n
\end{array}\right)b^{n}$
$\displaystyle \left(\begin{array}{l}
n\\
r
\end{array}\right)=\frac{n!}{r!(n-r)!},$
$n!=n(n-1)\cdot...\cdot 2\cdot 1,$
$ 0!=1$
--------
$...$In all the terms, exponents of a and b add to n...
In $(a+b)^{18}$
the middle term in the expansion is where exponents of a and b are 9.
So for $(x^{2}+1)^{18}$ this term is
$\left(\begin{array}{l}
18\\
9
\end{array}\right)(x^{2})^{9}1^{9}=$
$=\displaystyle \frac{18\cdot 17\cdot 16\cdot 15\cdot 14\cdot 13\cdot 12\cdot 11\cdot 10}{1\cdot 2\cdot 3\cdot 4\cdot 5\cdot 6\cdot 7\cdot 8\cdot 9}\cdot x^{18}$
$=48,620x^{18}$
