One more integral problem: integral [-1, 2] 1/4-x dx

[ricecrispie]

Integrate:

1/4-x

The antiderivative of this is: (I assume)

ln|4-x| then our upper limit was -1 and lower 2

So we have to say -ln|4-x| to get our upper limit to 2 and lower to -1

However when I sub in the values, I get -ln|2/5| and our answer sheet says it's ln|2/5|

Where am I going wrong?

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[HallsofIvy]

Integrate:

1/4-x

The antiderivative of this is: (I assume)

Never, ever "assume"! But I, unfortunately, have to assume that you mean 1/(4- x). What you wrote has anti-derivative (1/4)x- (1/2)x^2+ C.

ln|4-x|

No! The derivative of that would be 1/(4- x) times the derivative of 4- x which is -1: -1/(4- x)= 1/(x- 4).
The correct anti-derivative is -ln|4- x|+ C (which is the same as -ln|x- 4|+ C).

then our upper limit was -1 and lower 2

So we have to say -ln|4-x| to get our upper limit to 2 and lower to -1

You swapped the sign in order to swap the upper and lower limits but you had the wrong sign to begin with! The correct integral is (-ln(4- (-1)))- (-ln(4-2))= -ln(5)+ ln(2)= ln(2/5). (Since 2/5 is positive you don't need the absolute value signs.)

However when I sub in the values, I get -ln|2/5| and our answer sheet says it's ln|2/5|

Where am I going wrong?
You had the wrong sign on the anti- derivative.

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