 # Solving <msubsup> &#x222B;<!-- ∫ --> <mrow class="MJX-TeXAtom-ORD"> x uplakanimkk 2022-07-05 Answered
Solving ${\int }_{{x}_{1}}^{{x}_{2}}\frac{dx}{\sqrt{E-{U}_{0}\mathrm{tan}\left(ax{\right)}^{2}}}$
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Well, we are trying to find the following integral:

Let's substitute $\text{u}=\text{n}x$, this leads to:

Let's substitute $\text{s}=\mathrm{sin}\left(\text{u}\right)$, this leads to:

Let's substitute $\text{w}=\text{s}\cdot \sqrt{\frac{\beta }{\alpha }-1}$, this leads to:

This is a very standard integral, which gives:
$\begin{array}{}\text{(5)}& {\mathcal{I}}_{\text{n}}\left(\alpha ,\beta \right)=\frac{1}{\text{n}\sqrt{\beta -\alpha }}\cdot \mathrm{ln}|\text{w}+\sqrt{{\text{w}}^{2}+1}|+\text{C}\end{array}$
So, we end up with:
$\begin{array}{}\text{(6)}& {\mathcal{I}}_{\text{n}}\left(\alpha ,\beta \right)=\frac{1}{\text{n}\sqrt{\beta -\alpha }}\cdot \mathrm{ln}|\mathrm{sin}\left(\text{n}x\right)\cdot \sqrt{\frac{\beta }{\alpha }-1}+\sqrt{{\mathrm{sin}}^{2}\left(\text{n}x\right)\cdot |\frac{\beta }{\alpha }-1|+1}|+\text{C}\end{array}$

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