Indefinite integrals (or antiderivatives) are really just backward differentiation. Therefore, the indefinite integral of eˣ is eˣ+c, the indefinite integral of 1/x is ln(x)+c, the indefinite integral of sin(x) is -cos(x)+c, and the indefinite integral of cos(x) is sin(x)+c.
u-substitution is an extremely useful technique. Harnessing the power of the chain rule, it allows us to define a new variable (common denoted by the letter u) as a function of x, and obtain a new expression which is (hopefully) easier to integrate.
Certain ideas in physics require the prior knowledge of integration. The big idea of integral calculus is the calculation of the area under a curve using integrals. Let's do a fundamental course of integration.