Definition: An antiderivative of f is a function F whose derivative is f; every function has a whole family of antiderivatives differing by a constant.
What is an antiderivative?
If F'(x) = f(x), then F is an antiderivative of f. The family of all of them is the indefinite integral, written ∫ f(x) dx = F(x) + C. The constant C appears because constants vanish when you differentiate.
How do you find one?
Reverse the power rule: raise the exponent by one, then divide by the new exponent.
Worked example: find F with F'(x) = 8x3 – 5 and F(1) = 7. Antiderivative: F(x) = 2x4 – 5x + C. Use F(1) = 2 – 5 + C = 7, so C = 10.
The answer is F(x) = 2x4 – 5x + 10.
Check by differentiating: 8x3 – 5.
What goes wrong?
Dropping the +C, or applying the rule to x-1, which would divide by zero. The antiderivative of 1/x is ln|x|. Mathbot always differentiates the result as a quick check.