Question

Question attached as screenshot belowPlease help me with my calculus homework I am paying close attention

Answer 1

Given:

$\int ^1_{-1}\frac{1}{x^2+1}dx$

To evaluate:

As we know,

$\int ^{}_{}\frac{1}{x^2+1^{}}dx=\tan ^{-1}(x)+C$

So, we have

$\begin{gathered} \int ^1_{-1}\frac{1}{x^2+1}dx=\tan ^{-1}(1)-\tan ^{-1}(-1) \\ =\frac{\pi}{4}-(-\frac{\pi}{4}) \\ =\frac{\pi}{4}+\frac{\pi}{4} \\ =\frac{2\pi}{4} \\ =\frac{\pi}{2} \end{gathered}$

Hence, the answer is,

$\frac{\pi}{2}$