How Is Trigonometric Substitution Used in Solving Hyperbolic Functions?

[nameVoid]

<br /> 9x^2-4y^2=36<br />
<br /> \frac{x^2}{4}-\frac{y^2}{9}=1<br />
<br /> y=\frac{3}{2}\sqrt{x^2-4}<br />
<br /> 3\int_{2}^{3}\sqrt{x^2-4}dx<br />
<br /> x=2sect<br />
<br /> dx=2secttant<br />
<br /> 12\int_{a}^{b}tan^2tsectdt<br />
<br /> 12\int_{a}^{b}(sec^2t-1)(sect)dt<br />
<br /> 12\int sec^3tdt-12\int sectdt<br />
<br /> 6\int secttant-6\int ln|sect+tant|<br />
<br /> sect=\frac{x}{2}<br />
<br /> tant=\frac{\sqrt{x^2-4}}{2}<br />
<br /> \frac{3x\sqrt{x^2-4}}{2}-6ln|\frac{x+\sqrt{x^2-4}}{2}| [2,3]<br />
<br /> \frac{9\sqrt{5}}{2}-6ln|3+\sqrt{5}|+C<br />

 

[vela]

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