∫? x√x+5dx

Sure, here is the step-by-step solution to the integral ∫x√(x+5)dx:

Step 1: Use substitution method. Let u = x + 5. Then, du = dx and x = u - 5.

Step 2: Substitute x and dx in the integral. We get ∫(u-5)√udu.

Step 3: Distribute the square root of u inside the parenthesis. We get ∫(u√u - 5√u)du.

Step 4: Separate the integral into two parts. We get ∫u√udu - ∫5√udu.

Step 5: Solve the first integral ∫u√udu. Let v = u^2, then dv = 2udu. The integral becomes 1/2 ∫v^(1/2)dv = 1/2 * 2/3 * v^(3/2) = 1/3 * u^(3/2).

Step 6: Solve the second integral ∫5√udu. This is a standard integral and its solution is 10/3 * u^(3/2).

Step 7: Substitute u back into the solution. We get 1/3 * (x+5)^(3/2) - 10/3 * (x+5)^(3/2) + C, where C is the constant of integration.

So, the solution to the integral ∫x√(x+5)dx is 1/3 * (x+5)^(3/2) - 10/3 * (x+5)^(3/2) + C.