Solve using Integration by Substitution :

∫√x2-a2/x dx
∫√c2-a dx
∫ ln (√y + 3) dy
∫(5y+4) dy/√5y-1

To solve the given integral using integration by substitution, we follow these steps:

1. Let's start with the first integral:
∫√(x^2 - a^2)/x dx
To simplify the notation, let's represent (x^2 - a^2) as u.
u = x^2 - a^2
Taking the derivative of both sides with respect to x, we have:
du/dx = 2x
dx = du/(2x)

2. Now, substitute the values of u and dx into the integral:
∫√u/(x)(dx) = ∫√(u)/(x)(du/(2x))

3. Simplify the expression:
∫√(u)/(x)(du/(2x))
= (1/2)∫(√u/√x)(1/√x)(du)
= (1/2)∫(√(u)/x^(3/2))du

4. We know that √u = √(x^2 - a^2). So, we can rewrite the integral as:
= (1/2)∫(√(x^2 - a^2)/x^(3/2))du

5. Now, we can solve the integral:
= (1/2)∫(√(x^2 - a^2)/x^(3/2))du
= (1/2)∫(u^(-1/2))du
= (1/2) * 2/u^(1/2) + C
= √(x^2 - a^2)/x + C

Therefore, the solution to the first integral is √(x^2 - a^2)/x + C.

Follow the same steps for the other integrals:

1. For the second integral:
∫√(c^2 - a) dx

Let u = c^2 - a
Taking the derivative, du/dx = dx

Substituting the values of u and dx into the integral:
∫√u du

Integrating simplifies the expression to:
(2/3)(u^(3/2)) + C

Substitute the value of u we initially set:
(2/3)((c^2 - a)^(3/2)) + C

2. For the third integral:
∫ln(√y + 3) dy

Let u = √y + 3
Taking the derivative, du/dy = (1/2)(1/√y)

Rearranging for dy:
dy = 2√y du

Substitute the values of u and dy into the integral:
∫ln(u)(2√y)(du)

Integrating simplifies the expression to:
(2/3)(u^3)(ln(u)) + C

Substitute the value of u we initially set:
(2/3)((√y + 3)^3)(ln(√y + 3)) + C

3. For the fourth integral:
∫(5y + 4)/(√(5y) - 1) dy

Let u = √(5y) - 1
Taking the derivative, du/dy = (5/2)(1/√(5y))

Rearranging for dy:
dy = (2/5)√(5y) du

Substitute the values of u and dy into the integral:
∫(5y + 4)(2/5)(√(5y))(du)

Simplifying the expression:
(2/5) * ∫((5y + 4)√(5y)) du

This is beyond the scope of the Integral by substitution method, and further techniques like integration by parts may be required to solve it.