Integrating dx/sqrt(x^2+r^2) - Step-by-Step Guide

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In summary, to integrate the expression dx/sqrt(x^2+r^2), one can make the substitution x=ru and simplify to get du/sqrt(u^2+1). If this is still not recognizable, another potential substitution is x=r*sinh(u). Alternatively, one can try the trigonometric substitution x=r*tan(u). After trying the substitution x=r*tan(u), the desired solution can be obtained.
  • #1
omer21
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how can i integrate this expression

dx/sqrt(x^2+r^2)

in books i found just the answer,but i need the solution step by step

can you help me?
 
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  • #2
Now, let x=ru, where u is your new variable.

Try and simplify after this substitution of variables what you get as your integrand!
 
  • #3
after simplifying i got this

du/sqrt(u^2+1)

but still i can not see the solution
 
  • #4
If you don't recognize that integral, another useful substitution might be: x = r * sinh(u). Or, if you're unfamiliar with the hyperbolic functions, try the trigonometric substitution: x = r * tan(u).
 
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  • #5
hyperbolic functions are a little complicated for me so i will try x=r*tan(u)
 
  • #6
That's the more difficult substitution, but alright!
 
  • #7
i tried x=r.tan(u) substitution and i got what i want.
thanks...
 

1. What is the formula for integrating dx/sqrt(x^2+r^2)?

The formula for integrating dx/sqrt(x^2+r^2) is ln|x+sqrt(x^2+r^2)|+C.

2. How do I solve an integral with the form dx/sqrt(x^2+r^2)?

To solve an integral with the form dx/sqrt(x^2+r^2), you can use the substitution method. Let u = x^2+r^2 and du = 2x dx. The integral then becomes 1/2 * ∫(du/u^(1/2)). This can be solved using the power rule for integration, and then substitute back in for u.

3. Can you show me a step-by-step guide for integrating dx/sqrt(x^2+r^2)?

Yes, here is a step-by-step guide for integrating dx/sqrt(x^2+r^2):1. Use the substitution method to replace x^2+r^2 with u.2. Solve for dx in terms of du.3. Substitute dx and u into the original integral.4. Simplify the integral and solve using the power rule for integration.5. Substitute back in for u to get the final solution.

4. What is the significance of the constant r in the integral dx/sqrt(x^2+r^2)?

The constant r represents the radius of the circle in the denominator of the integral. It can also be thought of as the distance from the origin to the center of the circle.

5. Are there any special cases to consider when integrating dx/sqrt(x^2+r^2)?

Yes, if r = 0, then the integral becomes dx/sqrt(x^2+0^2) = dx/x = ln|x|+C. If r < 0, then the integral is undefined as the square root of a negative number is imaginary. Also, if the integral includes limits of integration, you may need to split the integral into multiple parts depending on the interval of the function.

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