Question

qual é a função f cuja derivada é dada por f(x) = 2e^2x - 3 e f(0) = 1?

Answer 1

$\displaystyle f'(x)=2e^{2x}-3\implies f(x)=\int f'(x)\,dx=\int2e^{2x}-3\,dx\implies\\\\ 2\int e^{2x}\,dx-\int 3\,dx\implies \ \ \ |u=2x\implies \frac{du}{dx}=2\implies dx=\frac{du}{2}\\\\ 2\int \frac{e^{u}\,du}{2}-\int3\,dx=2\left(\frac{e^{u}}{2}\right)-3x+C=2e^{u}-3x+C\implies\\\\ u=2x\implies \boxed{e^{2x}-3x+C=f(x)}$
$\displaystyle f(x)=e^{2x}-3x+C\ |\ f(0)=1\implies f(0)=e^{2\cdot0}-3\cdot0+C=1\\\\e^{0}-0+C=1\implies 1+C=1\implies C=0\\\\ A~funcao~primitiva~e:\ \boxed{f(x)=e^{2x}-3x}$