Integration: watch it solved step by step

Find the antiderivative, then prove it by differentiating back. Level: A-level. Type any problem of this kind and the Lab works it out and animates every move, with a plain-words explanation of why.

Worked example: sin(x)

1. The integral. Integration is the reverse of differentiation: we look for a function F(x) whose derivative is f(x). Then f(x)\,dx = F(x) + C.
2. Find an antiderivative. The engine finds F(x) by recognising standard patterns (powers, sin, cos, e^x, 1/x, products via parts, substitution). It does not narrate which pattern it used, so we will not pretend to. Instead we prove the answer by differentiating it back.
3. Add the constant. Differentiating a constant gives 0, so any constant could have been there. We write + C.
4. Prove it: differentiate F. Differentiate F(x) and see whether we get back the original function. At x = 1.3: F'(1.3) = 0.9636 and F was checked this way at 52 points.

Answer: -cos(x) + C

Practise integration with problems the Lab marks for you.