Is $\tan^{-1}(-1) = 3\pi /4$ or $=7\pi /4$? I understand they're both valid solutions, but what about places where the value is added/subtracted?

Question

For example: calculate $\int^4_2 \tan^{-1} x \, dx$

If $\tan^{-1}(-1) = 3\pi /4$, then the final answer is $\frac{-\pi}{2}$. But if $\tan^{-1}(-1)= 7\pi /4$, then the final answer would be $\frac{3}{-4\pi}$. Which is it?

Answer 1

The value of $\arctan$ is defined to lie within $(-\pi/2,\pi/2)$. Therefore the value should be $-\pi/4$.