instructions: show your work and explain your reasoning. write neatly. keep your eyes on your own paper. 1…

Explanation:

Step1: Recall the antiderivative

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

Step2: Apply the fundamental theorem of calculus

$\int_{0}^{1}\frac{1}{x^{2}+1}dx=\left[\arctan(x)\right]_{0}^{1}=\arctan(1)-\arctan(0)$.

Step3: Evaluate the arctangent values

We know that $\arctan(1)=\frac{\pi}{4}$ and $\arctan(0) = 0$. So $\int_{0}^{1}\frac{1}{x^{2}+1}dx=\frac{\pi}{4}$.

Step4: Compare with 3

Since $\frac{\pi}{4}\approx 0.785$ and $0.785<3$.

Answer:

The value of $\int_{0}^{1}\frac{1}{x^{2}+1}dx$ is $\frac{\pi}{4}$, and it is less than 3.