Evaluate the integral S x/(sqrt(3x-1))dx

Use U-Substitution method.

s=xsqrt

Let u^2 = 3x-1

2u du = 3 dx
and the integrand then becomes

(u^2+1)/u * 2u/3 du = 2/3 (u^2+1) du

Integrate that and then change back to x.

To evaluate the integral ∫x/√(3x-1) dx using the u-substitution method, follow these steps:

Step 1: Choose a substitution expression.
Let's choose u = 3x - 1.

Step 2: Find du/dx and solve for dx.
Differentiate both sides of the substitution expression with respect to x.
du/dx = 3
Solve for dx by rearranging the equation to isolate dx.
dx = du/3

Step 3: Rewrite the integral in terms of u.
Substitute the expression for dx and the expression for u into the original integral.
∫(x/√(3x-1)) dx = ∫(x/√u) (du/3)

Step 4: Simplify the integral.
Rearrange and simplify the expression obtained in step 3.
(1/3) ∫(x/√u) du

Step 5: Use u substitution to solve the integral.
Let's substitute u back into the integral obtained in step 4.
(1/3) ∫(x/√u) du = (1/3) ∫(x/u^(1/2)) du

Step 6: Evaluate the integral.
Integrate the expression with respect to u.
(1/3) ∫(x/u^(1/2)) du = (1/3) * 2x * u^(1/2) + C
= (2x/3) √u + C

Step 7: Substitute back in the original variable.
Replace the expression for u with the original expression for x.
(2x/3) √u + C = (2x/3) √(3x - 1) + C

Therefore, the solution to the integral ∫x/√(3x-1) dx using the u-substitution method is: (2x/3) √(3x - 1) + C, where C represents the constant of integration.

To evaluate the integral S x/(sqrt(3x-1))dx using the U-Substitution method, we start by making a substitution:

Let u = 3x - 1

To find du/dx, we differentiate both sides with respect to x:

du/dx = 3

Solving for dx, we have:

dx = du/3

Now, substitute the values of u and dx in terms of du in the integral:

S x/(sqrt(3x-1))dx = S (u+1)/(sqrt(u)) * (du/3)

Next, we simplify the expression within the integral:

= (1/3) * S (u+1)(u^(-1/2)) du

Now, we can expand the expression:

= (1/3) * S u^(1/2) + u^(-1/2) du

Integrate each term separately:

= (1/3) * [2/3 * u^(3/2) + 2 * u^(1/2)] + C

Finally, substitute the original variable back in terms of u:

= (2/9) * (3x - 1)^(3/2) + (2/3) * (3x - 1)^(1/2) + C

Therefore, the integral of x/(sqrt(3x-1))dx using the U-Substitution method is:

(2/9) * (3x - 1)^(3/2) + (2/3) * (3x - 1)^(1/2) + C