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

the enthusiast's antiderivative is incorrect

Explanation:

Answer 2

`(3x^2+1)/(2x)=\frac{3x}2+\frac1{2x}`

Integrating this gives

`\frac{3x^2}4+\frac12\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:

`(\mathrm d)/(\mathrm dx)\left((x^3+x)/(x^2)+C\right)=(\mathrm d)/(\mathrm dx)\left(x+\frac1x\right)=1-\frac1{x^2}=(x^2-1)/(x^2)`

but this is not the same as the original integrand.