I A troubling definite integral

1. Aug 8, 2016

Prasun-rick

• Member warned that homework questions must be posted in the HW & CW sections
∫|x^3-x|dx with limits from -1 to 2.Can anyone kindly show the step wise solution or suggest how to proceed .Thanks in advance ​

2. Aug 8, 2016

Krylov

Can you tell us first what makes it troubling for you?

3. Aug 8, 2016

BvU

First step: make a drawing !

4. Aug 8, 2016

Prasun-rick

The modulus sign! I In which limit I have to take positive and negative in the modulus I could not understand!

5. Aug 8, 2016

Prasun-rick

Well I am not able to draw precisely and that is where the problem lies !

6. Aug 8, 2016

BvU

Doesn't have to be precise. Simply investigate the function a little.
Where it comes from on the left, where it goes through zero and where it goes to on the right.
Perhaps there is even a little symmetry to explore and make good use of ...

7. Aug 8, 2016

Prasun-rick

well kindly please tell me the terminal points !! I found it as -1,0 and 1 am I correct??

8. Aug 8, 2016

BvU

I don't know what terminal points are but the zero points of $\ \ x^3-x = x(x+1)(x-1) \ \$ are indeed -1, 0, 1 .

Between -1 and 0 two out of three factors are < 0 and the other is > 0, so the expression is positive.
Between 0 and 1 ...
There is antisymmetry : $\ \ (-x)^3 -(-x) = -(x^3 - x)$, so ...

9. Aug 8, 2016

Prasun-rick

Thanks I finally got that !!Thank you very much !!

10. Aug 15, 2016

chiro

Hey Prasun-rick.

You will have to split up the function within the absolute value function so that it is positive and negative and create separate integrals for each section.

Just remember that you define |x| = x if x > 0 and -x if x < 0 (and 0 if x = 0).