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Can anyone help me with this? I used completing the square but do not know how to continue? Thanks

9.. Gaussian Integral

The following definite integration is particularly relevant in the subject of quantum mechanics (QM). (It always pops out, so it is worthwhile to memorise this when you are doing QM)

$$\int_{-\infty}^\infty e^{-ax^2} ~ dx = \left ( \frac{\pi}{a} \right )^{1/2}$$


$$\int_{-\infty}^\infty e^{-4x^2 + 3x} ~ dx$$

Hint: Complete the square for the exponent, i.e. write $4x^2 - 3x = \left ( 2x - \frac{3}{4} \right )^2 - \left ( \frac{3}{4} \right )^2$.

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1) Write it as: $$e^{\left(\frac{3}{4}\right)^{2}}\int_{-\infty}^{\infty} e^{-\left(2x-\frac{3}{4}\right)^{2}}dx$$

2) Apply substitution $u=2x-\frac{3}{4}$

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