Single-variable calculus from first principles
An antiderivative of f is a function whose derivative is f; you're running differentiation in reverse. The FTC says this is exactly what you need to evaluate integrals, so being fluent at "un-differentiating" is the key skill of integration.
To differentiate xⁿ you dropped the exponent by one and multiplied by it. To antidifferentiate, do the opposite: raise the exponent by one and divide by the new exponent:
An antiderivative is an 'undo' button. Someone hands you a slope — a derivative — and asks which function it came from, so you reverse the gesture that produced it. Differentiating took a function and reported its slope; antidifferentiating presses undo and hands the original function back (give or take a constant the undo can't see).