Integrating tan^2x

Question

My question includes quite a number of complex equations, so I instead upload  word file.

Faizan Ahmed · Accepted Answer

Since tan(x) whole square is not equal to {tan(tanx)}^2 thats why the second one is absolutely wrong.

Ian Fowler · Answer

I think I see where you are coming from now. The problem is the 13x in the denominator.  Your Short cut way: int[ (2x+3)^12]dx = (1/13) (2x+3)^13 / 2 ---> divide by the inner derivative of 2x+3 which is 2 to undo                                                    the chain rule. Not x as you were doing   = (1/26) (2x+3)^13 + c  Check by taking the derivative: d/dx [ (1/26)(2x+3)^13] = (1/26) (13) [(2x+3)^12)] (2)  -------->  times by 2 as the result of the chain rule. = (26/26)(2x+3)^12 = (2x+3)^12  I should point out that this little short cut only works when the inner derivative is a constant - in this case 2. If you are integrating ( 2x^2 + 3x)^13 then you are SOL.

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