As you know we can solve definite integrals using the definition over limit (without using antiderivatives at all). So I what I ask is how to solve an indefinite integral like that? For example $$\int x^2 \, dx$$ , we all know the result is x^3/3 + c. But we got this by using differentation and Fundamental Theorem of Calculus. How can we calculate this without using them?
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$\begingroup$ Indefinite integrals are by definition anti-derivatives. There is no direct limit definition for them. $\endgroup$ – Paramanand Singh Jan 6 '19 at 16:43
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Divide the interval into n parts and use $\sum_{k=1}^n k^2 =n(n+1)(2n+1)/6 =n^3/3+$ smaller terms.
Take the limit.
This is how these integrals were originally found.
answered Jan 6 '19 at 16:24
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$\begingroup$ Can you write the exact limit expression? It seems easy this way but I need it to understand for real. $\endgroup$ – madscientist Jan 7 '19 at 18:15
