Help me please! Calculus - Theorem

Question

Given that the antiderivative of  is F(x) = tan^-1(x) + C, evaluate 

Options:

1. π
2. π/2
3. π/4
4. 0

Answer 1

I think this is saying that given:

Find

This would just then be tan^-1(x)|¹_-1 = tan^-1(1) - tan^-1(-1)

The domain typically used for arctan (tan^-1) is [-π/2, π/2] so tan^-1(1) = π/4 and tan^-1(-1) = -π/4 so tan^-1(1) - tan^-1(-1) = π/4 - -π/4 = π/2

Answer 2

Given:

F(x) = ∫ f(x) dx = 1 / (x²+1) = tan^-1(x) + C

Graph of tan^-1(x):

tan^-1(1) = π / 4

tan^-1(-1) = -π / 4

Anti-derivative from -1 to 1 is F(1) - F(-1) = (π / 4) - (-π / 4) = π / 2

Answer: π / 2