Evaluate : ∫x2√5x − 3dx

To solve the integral ∫x²√(5x - 3) dx, we can use the method of substitution. Here are the steps:

1. Let u = 5x - 3. Then, du = 5dx. We also need to express x² in terms of u. From the substitution, x = (u + 3)/5. So, x² = (u² + 6u + 9)/25.

2. Substitute these into the integral: ∫x²√(5x - 3) dx = ∫((u² + 6u + 9)/25) * √u * (1/5) du.

3. Simplify the integral: ∫(1/125)(u^(3/2) + 6u^(1/2) + 9u^(1/2)) du.

4. Now, integrate term by term: (1/125)[(2/5)u^(5/2) + (12/3)u^(3/2) + 18u^(1/2)] + C.

5. Substitute u = 5x - 3 back into the integral: (1/125)[(2/5)(5x - 3)^(5/2) + 4(5x - 3)^(3/2) + 18(5x - 3)^(1/2)] + C.

This is the antiderivative of the given function.