Evaluate the indefinite integral

Note: Use an upper-case "C" for the constant of integration.

integral sqrt(100x^2-400)/x dx

Ty

What a mess!

http://integrals.wolfram.com/index.jsp?expr=sqrt%28100x%5E2-400%29%2Fx&random=false

add C to the result

To evaluate the indefinite integral ∫ √(100x^2 - 400)/x dx, we can take the following steps:

Step 1: Simplify the expression inside the square root.
The expression inside the square root, 100x^2 - 400, can be further simplified by factoring out a common factor of 100:
100x^2 - 400 = 100(x^2 - 4)
Now we have:
∫ √(100(x^2 - 4))/x dx

Step 2: Simplify the square root by separating it into two parts.
∫ √(100(x^2 - 4))/x dx
= ∫ (10√(x^2 - 4))/x dx

Step 3: Apply the u-substitution.
Let u = x^2 - 4
Then, du = 2x dx
Rearranging this equation, we get:
x dx = (1/2) du

Step 4: Rewrite the integral with the new variable.
∫ (10√(u))/x dx
= 10∫ √(u)/x dx
= 10∫ √(u) * (1/2) du
= 5∫ √(u) du

Step 5: Evaluate the integral of √(u).
By substituting u = x^2 - 4 back in, we have:
5∫ √(x^2 - 4) du
Now, this integral can be evaluated using various techniques, such as trigonometric substitutions, integration by parts, or a combination of algebraic manipulations.

Unfortunately, this integral does not have a simple closed-form solution. It leads to elliptic integrals, which are more complex. Therefore, we cannot directly compute the result without resorting to numerical methods or approximations.

In summary, the indefinite integral ∫ √(100x^2 - 400)/x dx cannot be evaluated symbolically and requires more advanced techniques for computation.