Trigonometric substitution

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

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vela
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