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