Read it again any time, including while you work through the exercises.
Since you learned to differentiate, you know how to turn a function into its derivative. What if we played the reverse game? I give you f, and you look for a function F whose derivative is f. Such a function F is called an antiderivative of f. For instance, F(x) = x² is an antiderivative of f(x) = 2x, since (x²)′ = 2x.
First surprise: an antiderivative never comes alone. Since the derivative of a constant is zero, x² + 5, x² - 3 and x² + 100 are also antiderivatives of 2x. Two antiderivatives of the same function always differ by a constant: there are infinitely many, all parallel.
To find antiderivatives, read the derivative table backwards. The derivative of xⁿ⁺¹ is (n+1)xⁿ, so an antiderivative of xⁿ is xⁿ⁺¹/(n+1). Concretely: an antiderivative of x is x²/2, an antiderivative of x² is x³/3, an antiderivative of 3x² is x³. The checking reflex is immediate: differentiate your antiderivative, and you must land back on the original function.
What is it for? Computing exact areas! The integral of f from a to b, written ∫f(x)dx from a to b, measures the area under the curve of f between x = a and x = b (when f is positive). And the fundamental theorem of calculus gives the recipe: this area equals F(b) - F(a), where F is any antiderivative of f. The constant cancels in the subtraction, which is why the choice of antiderivative does not matter.
This link between slope and area, between derivative and integral, is one of the most beautiful results in all of mathematics. Two seemingly unrelated questions, and a single theory that answers both.