Prove that if $f$ is integrable on $[a ,b]$ then $|f(x)|$ is also integrable on $[a ,b]$ and $|∫_a^b f(x)\,\mathrm dx|$ ≤ $∫_a^b|f(x)|\,\mathrm dx$

Question

Prove that if $f$ is integrable on $[a ,b]$ then $|f(x)|$ is also integrable on $[a ,b]$ and $|∫_a^b f(x)\,\mathrm dx|$ ≤ $∫_a^b |f(x)|\,\mathrm dx$

Answer 1

Hints:

1. $\displaystyle\left|\int_a^bf(x)\,\mathrm dx\right|\leqslant\int_a^b\bigl|f(x)\bigr|\,\mathrm dx\iff-\int_a^b\bigl|f(x)\bigr|\,\mathrm dx\leqslant\int_a^bf(x)\,\mathrm dx\leqslant\int_a^b\bigl|f(x)\bigr|\,\mathrm dx$
2. $(\forall x\in[a,b]):-\bigl|f(x)\bigr|\leqslant f(x)\leqslant\bigl|f(x)\bigr|$