Integration using a TI-89 produces wrong value?

[vnaut]

Hi Physics Forum,

I've just started taking Diff EQ and have been using my calculator for most integrations. I've come across two problems that I hope have answers.

1) When I try the integral of (x+1) (the quantity! not just x+1), my calculator still spits out (x^2/2)+x while the real answer should be (-(x+1)^2)/2. Is there a way to make my calculator identify (x+1) as the quantity (or is it factor? I forgot the actual term) rather than just x+1?

2) When I integrate 3/(100+2*x) I'm getting 3/2 * ln(x+50). However, I'm 99% sure the answer should be 3/2 * ln(100+2x). What am I/the calculator doing wrong?

Any and all answers are appreciated, thanks so much!

 

[Borek]

$$\frac 3 {100+2x} = \frac 3 {2(50+x)}$$

 

[vnaut]

$$\frac 3 {100+2x} = \frac 3 {2(50+x)}$$

This is where I'm confused. Multiple integral tables say that that

the integral of 1/(ax+c) is (1/a) * ln(ax+c). This would yield 3/2 * ln(100+2*x) where as the integral of your answer would yield 3/2 * ln(50+x). So I'm left with two different answers to choose from.

Either 3/2 * ln(100+2*x) or 3/2 * ln(50+x). Very curious =x

 

[AlephZero]

In both your examples, the difference in the "answers" is just a constant, so you can think of it as part of the arbitrary constant in an indefinite integral.

If you are going to evaluate a definite integral, the arbitrary constant cancels out so it doesn't matter.

In the first example the difference is the constant 1/2.

In the second example, from the properties of logarithms,
log(100 +2x) = log(2(50 + x)) = log(50+x) + log 2.