Integration of x/(ax + b) is a mathematical operation that involves finding the antiderivative of the given function. The integral of x/(ax + b) can be calculated using a technique called u-substitution.

U-Substitution Method:

To evaluate the integral of x/(ax + b), we can use the following steps:

1. Identify the function: x/(ax + b)

2. Apply u-substitution:
- Let u = ax + b
- Differentiate both sides of the equation with respect to x, we get du/dx = a
- Rearrange the equation to solve for dx: dx = du/a

3. Substitute the value of u and dx in the integral:
∫(x/(ax + b)) dx = ∫(1/a) (u - b/a) du

4. Simplify the integral:
∫(1/a) (u - b/a) du = (1/a) ∫(u - b/a) du
= (1/a) ∫(u - (b/a)) du

Now, we can integrate each term separately using the power rule of integration.

5. Integrate the terms:
∫(u - (b/a)) du = (1/a) ∫u du - (1/a) ∫(b/a) du
= (1/a) (u^2/2) - (1/a) (b/a) u + C

6. Simplify the result:
(1/a) (u^2/2) - (1/a) (b/a) u + C
= (u^2/2a) - (b/2a^2) u + C

7. Substitute the value of u back in terms of x:
= ((ax + b)^2)/(2a^2) - (b/2a^2) (ax + b) + C

8. Further simplify the result, if necessary.

Final Result:

The integral of x/(ax + b) is ((ax + b)^2)/(2a^2) - (b/2a^2) (ax + b) + C, where C represents the constant of integration.