Integral of sqrt(4-x²)/x using trigonometric substitution

[shadow5449]

I've almost gotten through this one integral problem, but I've seem to have gotten stuck:

integral ((sqrt(4-x^2))/x) dx
i let x = 2sin(theta), and dx = 2cos(theta)d(theta)
sqrt(4-x^2) = 2cos(theta)
integral ((2cos(theta))/(2sin(theta)) * 2cosd(theta)
2 integral cos^2(theta)d(theta)/sin(theta)

That's as far as I got, any suggestions?

 

[whozum]

Sure, express [itex]cos^2(\theta) [/tex] in terms of the sin function, sort of like yo udid earlier. You're on the right track.[/itex]

 

[thepatientmental]

It looks like you're on the right track with your substitution of x = 2sin(theta) and dx = 2cos(theta)d(theta). To continue solving this integral, you can use the trigonometric identity cos^2(theta) = (1+cos(2theta))/2. This will allow you to rewrite the integral as:

2 integral (1+cos(2theta))/sin(theta) d(theta)

From here, you can use the trigonometric identity sin(theta) = 2sin(theta/2)cos(theta/2) to simplify the integral further. This will give you:

4 integral (1+cos(2theta))/(2sin(theta/2)cos(theta/2)) d(theta)

Now, you can use the substitution u = sin(theta/2) to simplify the integral even more. This will give you:

4 integral (1+cos(2theta))/(2u) du

From here, you can use the fact that cos(2theta) = 1-2sin^2(theta) to simplify the integral further. This will give you:

4 integral (1+1-2sin^2(theta))/(2u) du

Finally, you can use the substitution v = sin(theta) to solve the integral completely. This will give you:

4 integral (1+1-2v^2)/(2v) dv

Now, you can integrate each term separately and plug back in your original substitution to get the final answer. Remember to check your answer by taking the derivative to see if it matches the original integrand. Keep up the good work and don't get discouraged, solving integrals can be tricky but with practice and patience, you'll get the hang of it!