MathDB

For the function

f(x)=\int_0^x \frac{dt}{1+t^2}

, answer the questions as follows.
Note : Please solve the problems without using directly the formula

\int \frac{1}{1+x^2}\ dx=\tan^{-1}x +C

for Japanese High School students those who don't study arc sin x, arc cos x, arc tanx.
(1) Find

f(\sqrt{3})

(2) Find

\int_0^{\sqrt{3}} xf(x)\ dx

(3) Prove that for

x>0

f(x)+f\left(\frac{1}{x}\right)

is constant, then find the value.

887