Answer to Question #103227 in Calculus for Patrick Kariuki

Question #103227
Integrate dx/(25x²-4)½ using trigonometric substitution
1
Expert's answer
2020-02-25T11:23:34-0500

Substitution:

x=25secθx=\frac{2}{5}\sec{\theta}

Then dx=25sinθcos2θdθdx=\frac{2}{5}\frac{\sin{\theta}}{\cos^2{\theta}}d\theta

dx25x24=25sinθcos2θ4sec2θ4dθ=15sinθcos2θsec2θ1dθ=\int\frac{dx}{\sqrt{25x^2-4}}=\int\frac{2}{5}\frac{\sin{\theta}}{\cos^2{\theta}\sqrt{4\sec^2{\theta}-4}}d\theta=\frac{1}{5}\int\frac{\sin{\theta}}{\cos^2{\theta}\sqrt{\sec^2{\theta}-1}}d\theta=

=15sinθcos2θtanθdθ=15dθcosθ=15secθdθ=15lnsecθ+tanθ+C.http://www.math.com/tables/integrals/more/sec.htm=\frac{1}{5}\int\frac{\sin{\theta}}{\cos^2{\theta}\tan{\theta}}d\theta=\frac{1}{5}\int\frac{d\theta}{\cos{\theta}}=\frac{1}{5}\int\sec{\theta}d\theta=\frac{1}{5}\ln|\sec{\theta}+\tan{\theta}|+C.\\ http://www.math.com/tables/integrals/more/sec.htm

secθ=52x\sec{\theta}=\frac{5}{2}x

x2=4251cos2θx2425=425tan2θtan2θ=254x21tanθ=254x21x^2=\frac{4}{25}\frac{1}{\cos^2{\theta}}\\ x^2-\frac{4}{25}=\frac{4}{25}tan^2{\theta}\\ tan^2{\theta}=\frac{25}{4}x^2-1\\ tan{\theta}=\sqrt{\frac{25}{4}x^2-1}

We have

dx25x24=15ln52x+254x21+C.\int\frac{dx}{\sqrt{25x^2-4}}=\frac{1}{5}\ln|\frac{5}{2}x+\sqrt{\frac{25}{4}x^2-1}|+C.


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