We are tasked with proving that for an integrable function $f$, the following inequality holds:

$\left| \int f(x) \, dx \right| \leq \int |f(x)| \, dx$

Proof:

1. Definition of absolute value of an integral:

   The absolute value of the integral of $f(x)$ can be written as: $\left| \int f(x) \, dx \right|$ By the triangle inequality for integrals, we know that the absolute value of the integral is less than or equal to the integral of the absolute value of the function: $\left| \int f(x) \, dx \right| \leq \int |f(x)| \, dx$ This inequality follows from the fact that integrating $f(x)$ may involve cancellations when $f(x)$ takes both positive and negative values, while integrating $|f(x)|$ eliminates these cancellations by treating all values of $f(x)$ as positive.

2. When does equality occur?

   Equality in the inequality $\left| \int f(x) \, dx \right| = \int |f(x)| \, dx$ occurs if and only if $f(x)$ is either non-negative or non-positive almost everywhere. This means that either:

   • $f(x) \geq 0$ almost everywhere, or
   • $f(x) \leq 0$ almost everywhere.

   In other words, equality holds if $f(x)$ does not change sign (i.e., it is either non-negative or non-positive) throughout the domain of integration. In this case, there are no cancellations due to positive and negative contributions from $f(x)$, and the magnitude of the integral of $f(x)$ equals the integral of $|f(x)|$.

Conclusion:

We have shown that for an integrable function $f$, $\left| \int f(x) \, dx \right| \leq \int |f(x)| \, dx$ with equality if and only if $f(x)$ is non-negative or non-positive almost everywhere.

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Do you need further details or have any specific questions?

Here are some related questions for you to consider:

1. What is the geometric interpretation of this inequality for integrals?
2. How does this inequality relate to the Cauchy-Schwarz inequality in integration?
3. Can this inequality be extended to functions of multiple variables?
4. How would this inequality change if we integrated over a finite interval $[a, b]$?
5. Does the inequality hold for improper integrals? Why or why not?

Tip: Always check for conditions where functions might change sign, as these are often where interesting inequalities come into play!