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In the following integral I want to change the bounds from $(0, 2)$ to $(-1, 1)$:

$\displaystyle{\int_{0}^{2}(1+x)^3 dx}$

How do I change them? I know that a variable changing is needed, but don't know how to use it to change the bounds. Indeed my question is how do I calculate the formula of the new variable(for example t)

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You want to shift the interval of integration down by $1$, so use the change of variables $t=x-1$. So when $x=0$, $t=-1$, and when $x=2$, $t=1$. Thus when integrating with respect to $t$, you would integrate over the interval $(-1,1)$. Be careful to rewrite your integrand in terms of $t$ before solving.

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Of course the original question admits multiple answers. – Siminore Jun 17 '12 at 10:47

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