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# does the integral of (tanh2x)^3 have two answers? watch

1. The integrals of (tanh2x)^3:

1) (1/2)ln(cosh2x) - (1/4)(tanh2x)^2 +c

2) (1/2)ln(cosh2x) + (1/4)(sech2x)^2 +c

Since in the part that needs integrating there is: (sech^2(2x))tanh2x)

So I thought I could do: let y=sech^2(2x)

then dy/dx = 4sech2x(-sech2xtanh2x) = -4sech^2(2x)tanh2x

so I = -1/4(sech^2(2x))

and then I include this overall and get (2)

What do you think?

Thank you!

I wanted to post this in the maths forum.
2. Well , so they're actually the same answer; the difference is accounted for in the arbitrary constant of integration.
3. Let's imagine an integral does have two different solutions.

If you differentiate their difference... what do you get?
And what does that tell you?
4. Here's the two functions in red and green (assuming the constant is zero), and finally in brown at the bottom is the difference between the two.
Attached Images

5. Thank you all

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