compute ∫x/√4x+14dx

To compute the integral ∫x/√(4x+14) dx, we can use the method of substitution. Here are the steps:

1. Let u = 4x + 14. Then, du = 4 dx. We can also write x = (u - 14) / 4.

2. Substitute these into the integral: ∫(u - 14) / (4√u) * (1/4) du.

3. Simplify the integral: (1/16) ∫(u - 14) / √u du.

4. Break up the integral: (1/16) [∫u / √u du - ∫14 / √u du].

5. Simplify the integrals: (1/16) [∫u^(1/2) du - 14∫u^(-1/2) du].

6. Now, we can integrate: (1/16) [(2/3)u^(3/2) - 28u^(1/2)] + C.

7. Substitute u = 4x + 14 back in: (1/16) [(2/3)(4x + 14)^(3/2) - 28(4x + 14)^(1/2)] + C.

This is the result of the integral ∫x/√(4x+14) dx.