Mod is the problem?

Calculus Level 3

$\large \int \left| x \right|^3 \, dx = \ ?$

Note : $C$ is the arbitrary constant of integration.

None of these choices

- \dfrac{x^4}{4} + C

\dfrac{x^4}{4} + C

\dfrac{x^3}{3} + C

1 solution

Niranjan Khanderia
Sep 9, 2015

$\large ~~\int|x|^3dx = \begin{cases} \dfrac{x^4} 4 +C&&;&& x~\geq~ 0~~ \\ -\dfrac{x^4} 4 +C&&;&& x~\leq~ 0 \end{cases}$

It can't be negative. This function is positive for whatever x. Make the substitution $\ |x|= \sqrt{x^2}$ in the integral and you'll get $\ \frac{x^4}{4} + C$

Leonardo de Araujo - 5 years, 7 months ago

The function, not its integral is positive.

Niranjan Khanderia - 5 years, 7 months ago

Hello, can u explain why if x<0 then it is negative? I thought that it would be positive because if we put that on a graph, the curve will be above the x axis always. Thanks

Ivan Martinez - 5 years, 8 months ago

There are two possibilities for the function. $It~could~be~+x^3~then~integral~is~positive. \\ OR~it ~could~have~been~ - x^3, ~the~ integral ~is~ negative.\\ \text{This is how it is solved. Sorry did not see your comment early. }$

Niranjan Khanderia - 5 years, 7 months ago

Mod is the problem?

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