Question

If \(f\) is continuous on \([a,b]\) and \(F\) is any antiderivative of \(f\), which formula evaluates the definite integral?

Options

• \[\int_a^b f(x)\,dx=F'(b)-F'(a)\]

• \[\int_a^b f(x)\,dx=F(a)-F(b)\]

• \[\int_a^b f(x)\,dx=F(b)-F(a)\]

• \[\int_a^b f(x)\,dx=F(x)+C\]

Answer 1

The Second Fundamental Theorem gives \(\int_a^b f(x)\,dx=F(b)-F(a)\).
Here \(F\) is any antiderivative of \(f\), so \(F'(x)=f(x)\).