Question

Evaluate the integral

Answer 1

\[\int\limits\limits_{-1}^{4}||x^2+x-6|-6| dx\]

Answer 2

The double absolute values are tripping me up.

Answer 3

you may be able to change the integration if theres symmetry. sec

Answer 4

I can factor it but I don't see symmetry.

Answer 5

I mean like how would I even do the cases?

Answer 6

if you plug in -1 and 4, its always going to be negative, so we can simplify it to be
-x^2+x+36

Answer 7

Are you sure about that?

Answer 8

CUz don't we have to break it into cases?

Answer 9

you might be able to do that

Answer 10

No. I get the wrong answer.

Answer 11

what did you get

Answer 12

Should be 71/6.

Answer 13

got to split up the intervals and apply appropriate sign so that the function is positive

Answer 14

Yes I got that. I factored the innermost quadratic but not sure how to deal with the outermost one.

Answer 15

when is x^2 + x - 6 > 6 ?

Answer 16

you may solve below to get the numbers where the intgrand changes sign
\[||x^2+x-6|-6| =0\]

Answer 17

Did you mean greater than 0 pg? That's true for x<3 and x>2 .

Answer 18

x<-3 sorry.

Answer 19

if you're allowed, you may simply graph it

Answer 20

No I cannot graph this haha.

Answer 21

I propose you go with ganeshie and integrate according to the graph

Answer 22

is it hard to solve :

\[||x^2+x-6|-6| =0\]

?

@Dido525

Answer 23

||x^2+x-6|-6| =0

|x^2+x-6|-6 =0

|x^2+x-6| =6

Answer 24

you can eyeball the solutions :

x = -1, 3

Answer 25

What about the negative case?

Answer 26

Also don't we have to consider ++,+-, -+, +- ?

Answer 27

definitely, we need to consider all that and sketch the graph

Answer 28

||x^2+x-6|-6| =0

|x^2+x-6|-6 =0

|x^2+x-6| =6

x^2+x-6 = 6 or x^2+x-6 = -6

you can solve right ?

Answer 29

Yes I can solve that. I think I have it. Hol don.

Answer 30

take ur time

Answer 31

Okay I am pretty far. I got to:

\[\int\limits_{-1}^4||x^2+x-6|-6|dx=\int\limits_{-1}^2|x^2+x|dx+\int\limits_{2}^4|(x+4)(x-3)|dx\]

Answer 32

Aha! I got it :) .

Answer 33

THanks everyone!

Answer 34

Awesome ! that integral looks perfect!