Answer:
The integral of ∫x²√(5x-3)dx is (2/15) * (5x-3)^(3/2) + C, where C is the constant of integration.
Explanation:
To find the integral of ∫x²√(5x-3)dx, we can use a technique called u-substitution.
Let's begin by letting u = 5x-3. To find du, we can take the derivative of both sides with respect to x.
du/dx = 5
Next, we solve for dx by isolating it.
dx = du/5
Now, we can substitute these values back into the integral.
∫x²√(5x-3)dx = ∫x²√u(dx/5)
Simplifying further, we have:
(1/5) ∫x²√u du
To integrate this expression, we can use the power rule for integration.
The power rule states that the integral of x^n with respect to x is (x^(n+1))/(n+1), where n is not equal to -1.
Applying the power rule, we have:
(1/5) ∫x²√u du = (1/5) * (2/3) * (u^(3/2)) + C
Substituting u back in, we get:
(1/5) * (2/3) * (5x-3)^(3/2) + C
Therefore, the integral of ∫x²√(5x-3)dx is (2/15) * (5x-3)^(3/2) + C, where C is the constant of integration.