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What strategy is the quickest for solving the following integral? Note: this integral is generated by the need to determine the magnetic field at a point along the z-axis generated by a wire of length $R\sqrt{2}$ centered on the x-axis at position $x = \frac{R}{\sqrt{2}}$ from $y = -\frac{R}{\sqrt{2}}$ to $y = \frac{R}{\sqrt{2}}$ using the law of Biot-Savart.

$$B = \frac{\mu_0 I}{4\pi}(z\hat{x} + \frac{R}{\sqrt{2}}\hat{z})\int_{-\frac{R}{\sqrt{2}}}^{\frac{R}{\sqrt{2}}} \frac{dy}{(\frac{R^2}{2}+y^2+z^2)^{\frac{3}{2}}}$$

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Your first starting point is a table of integrals, as that's a well known integral with an easy solution.

$$\int{\frac{dy}{(a^2+y^2)^{\frac{3}{2}}}}=\frac{y}{a^2\sqrt{a^2+y^2}}$$

In this case for the purposes of integration set :

$$a^2 = \frac{R^2}{2}+z^2$$

After that you can work out $a$, use the limits of integration and you have your formula.

I will let you do the rest.

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