It might not mean what you think it means!
What is the "core" anyway?
The core consists of the deepest layers of muscles in the body. When visualizing the core of a human body, think about an apple core - it's the deepest part and runs along the axis. Core muscles include: the deep muscles of the spine, the deep layers of abdominal muscles (psoas and transverse abdominus), the muscles of the pelvic floor, the diaphragm, and the inner muscles of the ribcage (inner intercostals).
Myths About Core
Myth #1: Core strength is about getting a "6-pack" and/or a "flat tummy"
The muscle that creates the 6-pack appearance is the rectus abdominus muscle, the most superficial abdominal muscle (which makes it inherently not a core muscle). More often than not, the main role that an overly worked-out rectus abdominus plays in core strengthening is that it interferes; in the case of someone with a "cut" stomach, often the rectus needs to relax and soften in order to access the real core body.
Myth #2: I should do "sit-ups" to strengthen my core.
It's possible to do something that resembles a sit-up or a crunch that might fire up the core muscles, but more often than not, "traditional-fitness-sit-ups" tend to activate muscles (like the rectus abdominus) that are not part of the core. Many people who do a lot of sit-ups have to go through a process of un-doing all the unbalanced strength they accumulated which interferes with true core awakening.
Myth #3: If I go to pilates classes, it means I am definitely working my core.
It is possible, and common, to participate in exercises that are considered core-targeted in essence, but end up using muscles other than the core. REALLY using the core in core-targeted exercises requires skill and nuanced understanding of how to engage internally.
What do you mean by an "integrated" core?
In this context, integrated means that we aren't talking about a randomly strong core that is discreet and isolated. The core should be strong in the sense of being healthy, alive, and powerful in a way that is appropriate relative to the rest of the body. If someone has a very strong, contracted core and very weak limbs and outer-layer muscles, then in order for that person to cultivate an integrated core and an integrated body, he/she would need to focus on releasing and lengthening the core and strengthening the weaker non-core muscles.
Why work on the core body?
Protect your back!
An appropriately strong and healthy core will ensure that your back doesn't have to work overtime. As the core muscles get stronger, they also also lengthen, providing a sturdy structural support for the spinal column, preventing spinal compression and related ailments.
Diminish chronic tightness and muscular compensation patterns.
If some muscles are under-working, others will be over-working. This manifests differently in every person.
Changes in other layers of your being.
As you create a stable core, examine changes that might occur in your mental and emotional body and in other parts of your life. The state of the core is the physiological expression of something deep inside of you.