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Is it valid to prove that [tex]\int_{a}^{c}f(x)dx=\int_{a}^{b}f(x)dx+\int_{b}^{c}f(x)dx[/tex]

using the fundamental theorem of calculus (FTC)?! That is, would it be valid to do the following.

Let [itex]F[/itex] be an anti-derivative of [itex]f[/itex] in an interval [itex][a,c]=[a,b]\cup [b,c][/itex]. It follows then (from the FTC), that [tex]\int_{a}^{b}f(x)dx=F(b)-F(a)[/tex] and [tex]\int_{b}^{c}f(x)dx=F(c)-F(b)[/tex] As such, [tex]\int_{a}^{b}f(x)dx+\int_{b}^{c}f(x)dx= \left[F(b)-F(a)\right]+\left[F(c)-F(b)\right]=F(c)-F(a)=\int_{a}^{c}f(x)dx[/tex] where the last equality follows from the assumption that [itex]F[/itex] be an anti-derivative of [itex]f[/itex] in an interval [itex][a,c][/itex] and hence [tex]\int_{a}^{c}f(x)dx=F(c)-F(a)[/tex]

In a similar manner, is it valid to use the FTC to prove that [tex]\int_{a}^{b}f(x)dx=-\int_{b}^{a}f(x)dx[/tex]