Question

How to evaluate the integral

Answer 1

Remember
for F(x) is the antiderivitive of f(x)
$\int\limits^a_b {f(x)} \, dx =F(a)-F(b)$
so find the antiderivitive of ((x+1)^2)/x

if we expand we get (x^2+2x+1)/x which simplifies to x+2+(1/x)
the anti-deritivive of  x is (1/2)x^2
the antideritiveve of 2 is 2x
the antideritivieve of 1/x is ln|x|

F(x)=(1/2)x^2+2x+ln|x|+C

$\int\limits^2_1 { \frac{(x+1)^2}{x} } \, dx =F(2)-F(1)$

F(1)=(5/2)+ln1+C
F(2)=6+ln2+C

F(2)-F(1)=6+ln2+C-(5/2+ln1+C)
F(2)-F(1)=(7/2)+ln2
that is the answer
if you want is simplified or expanded it is about 4.1915