Geeky stuff.
Integrate the following function with respect to x.
∫ ((sec^2)x).tanx dx
Integrate the following function with respect to x.
∫ ((sec^2)x).tanx dx
You can now substitute (secx) as t and (secx.tanx dx) as dt, bringing it to the form
∫ t dt = (1/2)(t^2) + c = ((sec^2)x)/2 + c ...(1)
Or, alternatively, you could substitute (tanx) as t and ((sec^2)x dx) as dt, thus bringing it to the form
∫ t dt = (1/2)(t^2) + c = ((tan^2)x)/2 + c ...(2)
My question is precisely this:
Does that mean (1) and (2) represent the same function? My knowledge and a little research tell me that they don't. But then, if integration is supposed to be representative of the area bound by the function and the X-axis, should it not be ONE single function (never mind that some functions can be expressed in more than one form involving the same variable and that does NOT make them different functions at all)?
And that's all, folks. The comments section is open to geek-giri.
My answer is precisely this:
ReplyDeletefirst of all I can see that there is only one function that is f(x)= sec^x.tanx and others are just visibly different results obtained due to two paths taken to reach an acceptable solution.
the integrals in 1 and 2 are a part of trigonometrical identity ( sec^x = tan^x +c. so, when you apply the identity in eqn (1), you get (2) along with some constants( thats why we add that c anyway for we are unsure of the constant)and vice-versa.
Simply put,plot the graph of the original integrand ( sec^x tanx) and don't bother much about (1) or (2)). two different paths to the same destination and they both start from a common source. the area of both (1) and (2) are same. if they were different functions, it wouldn't have been possible under same limits.
would be awaiting for your follow up. interesting anyway.