Question

Derivatives have a chain rule, how would I do it to for an integral?

Answer 1

\[\int\limits_{ }^{ } (x^2+3)^3~~dx\]

Answer 2

I don't have to expand to integrate, right?
Just like when I derive such a function I can use chain.

Answer 3

@uri

Answer 4

The chain rule "counterpart" for integrals isn't really a rule, but rather a strategy. It's why you use substitutions, so as to tack on factors that happen to be derivatives of expressions you would substitute.

Simple example: Suppose \(f(x)=e^x\). If \(g(x)=2x\), then \(f(g(x))=e^{2x}\), and \(f'(g(x))\times g'(x)=2e^{2x}\), as per the chain rule.

In reverse: Given \(h(x)=2e^{2x}\), we want to compute \(\int h(x)~dx\), we would let \(\color{blue}{t=2x}\), then \(\color{red}{dt=2~dx}\)
\[\int \color{red}2e^{\color{blue}{2x}}~\color{red}{dx}=\int e^\color{blue}t~\color{red}{dt}=e^{\color{blue}t}+C=e^{2x}+C\]

Answer 5

Your particular integral is rather tricky... I don't think you can find a substitution that will work perfectly, at least not without doing more work than an expansion would require.