random integration question (1 Viewer)

[gamja]

how do you reconcile integrations with more than one answer? Is the difference accounted for with differing constant 'c's? Does that technically mean any answer for an integration is correct? argh

e.g.
when solving rev chain and using double angle. Clearly sin^2x/2 does not equal cos2x/4.
[edit: I forgot the negative for cos2x/4. i think my point still stands tho]

 

[yanujw]

how do you reconcile integrations with more than one answer? Is the difference accounted for with differing constant 'c's? Does that technically mean any answer for an integration is correct? argh

e.g. View attachment 38273
when solving rev chain and using double angle. Clearly sin^2x/2 does not equal cos2x/4.
[edit: I forgot the negative for cos2x/4. i think my point still stands tho]

They differ by a constant so they are both correct antiderivatives. To check without a graphing calculator, just differentiate and see if you get the original function.

 

[gazzaboy]

Don't forget you also have another answer if you use the "reverse chain rule" on the cosine function!

All three answers will differ by a constant (you can see this by using the double angle formulas for cos(2x)). This doesn't mean that any answer for an integral is correct because there are functions you can't get to by just adding a constant.