To solve the definite integral from 0 to 2|b| of (x divided by sqrt(x^2 + b^2)) dx, we can use the substitution method.
1. Let's start by substituting u = x^2 + b^2. This means du = 2x dx.
Rearranging this equation, we have dx = du / (2x).
2. Now let's substitute the expression for dx in terms of du and x into the integral:
∫ (x / sqrt(x^2 + b^2)) dx = ∫ ((1 / (2x)) * u^(-1/2)) du.
3. Simplifying the integrand further, we have:
(1 / 2) * ∫ (u^(-1/2) / x) du.
4. We can now evaluate the integral of u^(-1/2) with respect to u:
(1 / 2) * ∫ u^(-1/2) du = (1 / 2) * (2u^(1/2)) + C,
where C is the constant of integration.
5. Substituting back u = x^2 + b^2, we get:
(1 / 2) * (2u^(1/2)) + C = u^(1/2) + C = sqrt(x^2 + b^2) + C.
6. To find the definite integral between 0 and 2|b|, we evaluate the expression at the upper limit (2|b|) and subtract it from the expression evaluated at the lower limit (0):
[sqrt((2|b|)^2 + b^2) + C] - [sqrt((0)^2 + b^2) + C].
Note: The constant of integration cancels out because we are calculating the difference between two expressions.
So, the definite integral from 0 to 2|b| of (x / sqrt(x^2 + b^2)) dx is equal to:
sqrt((2|b|)^2 + b^2) - sqrt(b^2)