Question

determinar a função f tal que integral de f(x)dx=x^2+1/2cos(2x)+c

Answer 1

É só derivar. Acompanhe:

$\displaystyle\mathsf{ y= x^2+\frac{1}{2} \cos 2x + C } \\ \\ \\ \mathsf{y' = 2x + \frac{1}{2} \cdot (-\sin 2x \cdot 2)} \\ \\ \\ \mathsf{y' = 2x - \sin 2x}$

Verificando:

$\displaystyle \mathsf{\int 2x - \sin 2x \, \, dx} \\ \\ \\ \mathsf{2 \cdot \frac{x^2}{2}-\frac{1}{2} \cdot (-\cos 2x ) + C} \\ \\ \\ \mathsf{x^2+\frac{1}{2} \cos 2x + C}$