# Indefinite Integral Formula

There is a simple formula for calculating $$\frac{df(x)}{dx} u^n$$ where u is a function of x and n is a positive rational number: $$\frac{df(x)}{dx} u^n = nu^{n-1} \ast \frac{du}{dx}$$. Is there a similar formula for calculating $$\int u^n dx$$ where u is a function of x and n is a positive rational number? It would be extremely helpful if there was.

P.S. I realize that the formula for $$\frac{df(x)}{dx} u^n$$ can be derived using the chain rule, so I was wondering if maybe the chain rule can somehow be applied in reverse for this problem?

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dextercioby
Homework Helper
There is a simple formula for calculating $$\frac{df(x)}{dx} u^n$$ where u is a function of x and n is a positive rational number: $$\frac{df(x)}{dx} u^n = nu^{n-1} \ast \frac{du}{dx}$$. Is there a similar formula for calculating $$\int u^n dx$$ where u is a function of x and n is a positive rational number? It would be extremely helpful if there was.

P.S. I realize that the formula for $$\frac{df(x)}{dx} u^n$$ can be derived using the chain rule, so I was wondering if maybe the chain rule can somehow be applied in reverse for this problem?

Actually that "f(x)" should not be there. In the Leibniz notation the differentiation operator is simply $\frac{d}{dx}$.

As for the question itself, there's no general method for computing $\int u^{n}(x) {} dx$ for arbitrary "u(x)". In most cases, one can't find the antiderivative in terms of elemetary functions, even though "u(x)" may be elementary.

HallsofIvy
$$\int u^n \frac{du}{dx}dx= \int u^n du= \frac{1}{n+1}u^{n+1}$$
$$\int (ax+b)^n dx= \frac{1}{a(n+1)}(ax+b)^{n+1}$$