1. ## Integral

Let $f$be integrable in $[a,b].$Prove that:

$
\int_{a}^{b} |f| \leq{\displaystyle\int_{a}^{b}|f|}
$

Thanks

2. Originally Posted by osodud
Let $f$be integrable in $[a,b].$Prove that:

$
\int_{a}^{b} f| \leq{\displaystyle\int_{a}^{b}|f|}
$

Thanks
I don't understand what you mean.

3. Originally Posted by VonNemo19
I don't understand what you mean.
I have edited the expression.

4. Originally Posted by osodud
Let $f$be integrable in $[a,b].$Prove that:

$
\int_{a}^{b} |f| \leq{\displaystyle\int_{a}^{b}|f|}
$

Thanks
Something doesn't seem right here...do you possibly mean

$\int_{a}^{b} |f| \leq\left|\int_{a}^{b}f\right|$

or

$\left|\int_{a}^{b} f\right| \leq\int_{a}^{b}\left|f\right|$

5. Originally Posted by Chris L T521
Something doesn't seem right here...do you possibly mean

$\left|\int_{a}^{b} f\right| \leq\int_{a}^{b}\left|f\right|$
Yesˇˇ That one

6. Originally Posted by Chris L T521
$\left|\int_{a}^{b} f\right| \leq\int_{a}^{b}\left|f\right|$
Originally Posted by osodud
Yesˇˇ That one
First note that by the FTC, $\int_a^b f= F(b)-F(a)$.

Thus, $\left|\int_a^b f\right|=\left|F(b)-F(a)\right|$.

Applying the triangle inequality, we have

\begin{aligned}\left|F(b)-F(a)\right| &\leq \left|F(b)\right|+\left|F(a)\right|\\ &\leq F(a)+F(b)\\ &\leq F(a)-F(x_1)+F(x_2)-F(x_1)+\ldots-F(x_{k-1})+F(b)\\ &=-(F(x_1)-F(a))+(F(x_2)-F(x_1))+\ldots+(F(b)-F(x_{k-1}))\\ &=-\int_a^{x_1}f+\int_{x_1}^{x_2}f-\int_{x_2}^{x_3}+\ldots+\int_{x_{k-1}}^bf\\ &=\int_a^b\left|f\right|\end{aligned}

I think that's the way to approach it, but I'll let others chip in (or inform me I'm not correct)