Integration Questions

[frog1944]

Hi,

I have a few integration questions, based on the formatting of the answers for integration.

1. For , the answers had , whereas I had simplified the inverse trig to . Is one answer in a more preferred form? Or more correct?

2. When it comes to using trig substitutions, in particular when you end up square rooting something squared, in your answer do you need to briefly discuss the domain of that trig function? As the square root of a square is the absolute value.

3. When it comes to integrating functions that eventually result in a logarithm, do you need to include the absolute value signs all the time? If not, why not? E.g. or should it be

Thanks

[RuiAce]

1. First one feels safer. If you choose to expand out the sine via double angles you need to be extremely careful handling domain restrictions on x. Sometimes it's easy, other times it's not.

2. Specify a domain restriction at the very start so that you may take the positive root without problem.
sin: -π/2≤theta≤π/2
tan: -π/2<theta<π/2
sec: I forgot, but maybe 0<theta<π/2, π/2<theta<π

3. If it involves log, the only time you don't use absolute values is when the expression you're taking a log of is never negative. If it can be negative, you must use absolute values.

[frog1944]

Could you perhaps give an example of where you have to be really careful of the domains RuiAce?

[RuiAce]

Could you perhaps give an example of where you have to be really careful of the domains RuiAce?

Which is of course, the most unlikely thing ever but you can't just rule it out completely.

[jamonwindeyer]

Which is of course, the most unlikely thing ever but you can't just rule it out completely.

Ew

[RuiAce]

Ew

I've seen worse

[frog1944]

Which is of course, the most unlikely thing ever but you can't just rule it out completely.

I take your point. Though what if it was the sine of inverse sine? Would you still have the same issue?

[RuiAce]

I take your point. Though what if it was the sine of inverse sine? Would you still have the same issue?

I actually included these inverse trig formulas in my 4U Notes

Edit: That should not have said pi/2

[frog1944]

Thanks