The indefinite integral of the function (4x^2 + 2√x + 1)/(2x√x) is calculated as ∫(4x^2 + 2√x + 1)/(2x√x) dx. The solution simplifies to 4/3√(x^3) + ln(x) - 1/√x + C. The process involves breaking down the integral into manageable parts, specifically integrating terms like 2∫x^(1/2) dx, ∫x^(-1) dx, and 1/2∫x^(-3/2) dx. Verification of the solution can be done by taking the derivative of the result.
PREREQUISITESStudents studying calculus, particularly those focusing on integration techniques, as well as educators looking for examples of indefinite integrals involving rational functions.
It looks good. Take the derivative to check it.KMcFadden said:Homework Statement
∫▒〖(4x^2+2√x+1)/(2x√x) dx〗
Homework Equations
The Attempt at a Solution
∫▒〖(4x^2+2√x+1)/(2x√x) dx〗
∫▒((4x^2)/(2x^(3/2) )+(2√x)/(2x^(3/2) )+1/(2x^(3/2) ))dx
2∫▒x^(1/2) dx+∫▒〖x^(-1) dx〗+1/2 ∫▒x^(-3/2) dx
2×2/3 x^(3/2)+lnx+1/2×(-2) x^(-1/2)+C
4/3 x^(3/2)+lnx-x^(-1/2)+C
4/3 √(x^3 )+lnx-1/√x+C