Proving $\int_{a}^{b}f(x)dx=\int_a^{c}f(x)dx+\int_c^{b}f(x)dx$

Question

I need to show that $$\int_{a}^{b}f(x)dx=\int_a^{c}f(x)dx+\int_c^{b}f(x)dx$$ So far I've only thought of showing a graph and join both of the intervals. However I think I need something better than visual proof. Thanks.

Answer 1

$$ \int_a^{c}f(x)dx + \int_c^{b}f(x)dx = \bigg(F(c)-F(a) \bigg ) + \bigg(F(b)-F(c) \bigg ) = F(b)-F(a)=\int_a^{b}f(x)dx$$

Answer 2

If there exists an anti derivative of $f(x)$ then $\int_{a}^{b}f(x)dx = \phi(b) - \phi(a)$ , where $\phi(x)$ is the anti derivative of $f(x)$.

Similarly,

$\int_{a}^{c}f(x)dx = \phi(c) - \phi(a)$

And,

$\int_{c}^{b}f(x)dx = \phi(b) - \phi(c)$

Adding the above equations, we get,

$ \int_{a}^{c}f(x)dx + \int_{c}^{b}f(x)dx= \phi(b) - \phi(a)$

$ \int_{a}^{c}f(x)dx + \int_{c}^{b}f(x)dx= \int_{a}^{b}f(x)dx$