Integration of x^(3/2)/(2x-1)

To find the integration of the given expression, we can use the method of substitution. Let's break down the process step by step:

Step 1: Identify the function
The given expression is x^(3/2)/(2x-1). We need to find its antiderivative or integral.

Step 2: Simplify the function
We can simplify the given expression by dividing the numerator and denominator by x^(1/2). This gives us:
(x^(3/2))/(2x-1) = (x^(3/2))/(2x^(1/2) - 1/x^(1/2))

Step 3: Perform substitution
Let's substitute u = x^(1/2). This implies that du/dx = (1/2)x^(-1/2). Rearranging, we get dx = 2u du.

Step 4: Rewrite the integral
Using the substitution, we can rewrite the integral as:
∫(x^(3/2))/(2x-1) dx = ∫(u^2)/(2u^2 - 1) * 2u du

Step 5: Simplify and solve the integral
Now, we can simplify the integral further:
∫(u^2)/(2u^2 - 1) * 2u du = ∫(2u^3)/(2u^2 - 1) du

Step 6: Divide the numerator by the denominator
To make it easier to integrate, let's divide the numerator (2u^3) by the denominator (2u^2 - 1):
∫(2u^3)/(2u^2 - 1) du = ∫(u^3 * 2)/(u^2 * 2 - 1) du

Step 7: Apply the power rule for integration
Now, we can apply the power rule for integration, which states that ∫x^n dx = (1/(n+1)) * x^(n+1) + C, where C is the constant of integration.
Using this rule, we integrate the simplified expression:
∫(u^3 * 2)/(u^2 * 2 - 1) du = 2∫(u^3)/(u^2 * 2 - 1) du

Step 8: Perform further simplification
We can simplify the denominator (u^2 * 2 - 1) by expanding it:
2∫(u^3)/(u^2 * 2 - 1) du = 2∫(u^3)/(2u^2 - 1) du

Step 9: Apply the power rule for integration again
We can now apply the power rule for integration to the simplified expression:
2∫(u^3)/(2u^2 - 1) du = 2*(1/4)∫(2u^2 - 1 + 1)/(2u^2 - 1) du

Step 10: Evaluate the integral
By applying the power rule for integration, we can