A fellow calculus enthusiast was working through some practice problems.They come to you asking if the following problem is correct: ∫ [(3x^2 + 1) / 2x] dx = [(x^3 + x) / x^2] + cDetermine if they are correct. Explain how you know if they are correct without integrating the function (i.e. find another method using calculus).

Accepted Solution

[tex]\dfrac{3x^2+1}{2x}=\dfrac{3x}2+\dfrac1{2x}[/tex]Integrating this gives[tex]\dfrac{3x^2}4+\dfrac12\ln|x|+C[/tex]so the enthusiast's antiderivative is incorrect.Without integrating, you can show the enthusiast's solution is incorrect by taking the derivative:[tex]\dfrac{\mathrm d}{\mathrm dx}\left(\dfrac{x^3+x}{x^2}+C\right)=\dfrac{\mathrm d}{\mathrm dx}\left(x+\dfrac1x\right)=1-\dfrac1{x^2}=\dfrac{x^2-1}{x^2}[/tex]but this is not the same as the original integrand.