Stretch the function

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Probably the easiest way to evaluate this integral is to expand the bracket (later you will learn about other ways to do it). \[\int_{-1}^2 x^2+2x+1\,dx =\left[\frac{x^3}{3}+x^2+x\right]_{-1}^{2}=\left(\frac{8}{3}+4+2\right)-\left(-\frac{1}{3}+1-1\right)=9\]

If we want \[\int_{-1}^2 a(x+1)^2 \,dx = 1 \] then we need our integral to be \(9\) times smaller. We can do this by scaling (i.e. vertically stretching) our function by a factor of \(\dfrac{1}{9}\). So \(a=\dfrac{1}{9}\).