# Explain how to solve the following anti-derivative

I am trying to figure how to solve the following antiderivative.

$$\int (5x+3)^7 dx \\$$ I've seen the step-by-step solution by WolframAlpha however what they are doing in this part:

For the integrand $(5x+3)^7$, substitute $$u = 5x + 3 \\ du = 5 dx \\$$

Why are they derivating $u$ ?

Thanks.

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Are you familiar with integration by substitution? en.wikipedia.org/wiki/Integration_by_substitution –  anonymous Dec 2 '12 at 16:55
Yes, I am, I understand now, I had no simple example –  Francis Dec 2 '12 at 17:02

The general theorem is like this:

$$\int (f\circ g)(x)g'(x)dx=\int f(u)du \Leftrightarrow u=g(x).$$

It may help to note that in calculations, we have $u=\text{something}$ and $du=(\text{something})'dx$. This is because of the definition of the differential. That is, $dy=f'(x)dx$ if and only if $y=f(x)$. (The definition of the differential just comes from the "multiplying" of $dx$ to both sides in $\frac{dy}{dx}=f'(x)$.)

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Directly:

$$\int (5x+3)^7dx=\frac{1}{5}\int 5(5x+3)^7dx=\frac{1}{5}\int(5x+3)^7d(5x+3)=$$

$$=\frac{1}{5}\frac{(5x+3)^8}{8}+C=\frac{(5x+3)^8}{40}+C$$

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