Question

Determine f(1), se ∫_0^x▒〖f(t)dt=cosπx.〗 (Use o Teorema Fundamental do Cálculo: F(x)=f(x))

Answer 1

$\int\limits_{0}^{x}f(t)dt=cos(\pi x)$

Derivando os dois lados da equação:

$\frac{d}{dx}\int\limits_{0}^{x}f(t)dt=\frac{d}{dx}cos(\pi x)\\\\f(x)=\frac{d}{dx}cos(\pi x)\\\\f(x)=-sen(\pi x)\cdot\frac{d}{dx}(\pi x)\\\\\boxed{\boxed{f(x)=-\pi sen(\pi x)}}$

Então:

$f(1)=-\pi sen(\pi\cdot1)=-\pi sen(\pi)=-\pi\cdot0=0$