Find a formula for $f(x)$ if
\[
f^{\prime}(x)=\frac{4}{1+x^{2}}, \quad f(1)=0
\]
Use this formula to compute $f(0)$.
Give your answer as a decimal number with at least three decimal places.

Answer

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Answer

Final Answer: $f(0) = \boxed{-3.142}$

Steps

Step 1 ：We are given that the derivative of the function, $f'(x)$, is $\frac{4}{1+x^{2}}$.

Step 2 ：To find the original function, $f(x)$, we need to find the antiderivative of $f'(x)$.

Step 3 ：The antiderivative of $\frac{4}{1+x^{2}}$ is $4\arctan(x)$.

Step 4 ：We also know that $f(1) = 0$, which we can use to find the constant of integration.

Step 5 ：After finding the function $f(x)$, we substitute $x = 0$ into the function to find $f(0)$.

Step 6 ：Final Answer: $f(0) = \boxed{-3.142}$