>>14430086>To integrate the following x2 + 4x + 2 gives 2x + 8That's differentiating, not integrating.

>Why does this work?It can be shown by the definition of the derivative if you know how to do the binomial expansion. Alternatively, it pops out of the product rule with induction. You should probably watch 3blue1brown's essence of calculus series. He's unironically pretty good at providing intuition behind mathematical concepts.

>can anyone give me a more solid example of where simple integration would be useful in real life?Depends on what you mean by useful in real life. If you are asking if you'll ever use integration when shopping for groceries or paying for taxes, no. If you mean usefulness in other fields, I guess one example you're probably unaware of is probability/statistics. Integrals are how you calculate probabilities on continuous distributions. The most fundamental probability distribution (Gaussian) especially so, as its CDF can only be expressed by an integral.