# quirk involving trig substitution?

I have reduced the following trig identity to the following which is correct.

$$\int \cos^2(x)\tan^3(x)dx = \int \tan(x) - \sin(x)\cos(x)dx$$

However this next step changes the value of my equation.. i.e False

$$\int \cos^2(x)\tan^3(x)dx = \int \tan(x) dx - \int\sin(x)\cos(x)dx$$

Why can i not do this?

proof below http://www.wolframalpha.com/input/?i=integral%28cos^2x*tan^3x%29+%3D+integral%28tanx%29+-+%28sinxcosx%29

http://www.wolframalpha.com/input/?i=integral%28cos%5E2x%2Atan%5E3x%29+%3D+integral%28tanx%29+-+integral+%28sinxcosx%29

• What? Wolfram gave the exact same result for both. – mattos Feb 25 '15 at 12:45
• oops, both links were pointing to the same page – Arden Feb 25 '15 at 13:57
• I can't follow the first link - but to comment generally, differences in the apparent form of indefinite integrals are quite often down to hidden differences in the constant of integration. – Mark Bennet Feb 25 '15 at 14:01

• i do not understand your answer completley, i have used similar properties where $$\int x−x^2 dx = \int x dx − \int x^2 dx$$ why is this property not working in this example? Are you suggesting because one willl have 2 constanst while the other wil only have one? – Arden Feb 25 '15 at 15:44
• There is a ${}+C$ for an indefinite integral. Mathematica chose its constants differently in the two cases. – GEdgar Feb 25 '15 at 18:52