Question

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] + c

Determine if they are correct. Explain how you know if they are correct without integrating the function (i.e. find another method using calculus).

Answer 1

$\dfrac{3x^2+1}{2x}=\dfrac{3x}2+\dfrac1{2x}$

Integrating this gives

$\dfrac{3x^2}4+\dfrac12\ln|x|+C$

so the enthusiast's antiderivative is incorrect.

Without integrating, you can show the enthusiast's solution is incorrect by taking the derivative:

$\dfrac{\mathrm d}{\mathrm dx}\left(\dfrac{x^3+x}{x^2}+C\right)=\dfrac{\mathrm d}{\mathrm dx}\left(x+\dfrac1x\right)=1-\dfrac1{x^2}=\dfrac{x^2-1}{x^2}$

but this is not the same as the original integrand.

Answer 2

Answer:

the enthusiast's antiderivative is incorrect

Step-by-step explanation: