Antiderivative is an operation which is opposite of derivation operation that means antiderivative calculates the integral of a derivative.
Suppose we have a function f(x), its derivative is g(x) means d(f(x))= g(x) than antiderivative of g(x) is f(x) that is ∫ g(x) dx = f(x) + c dx.
So we get f(x) as antiderivative of g(x) is a constant.
We have understood the meaning of antiderivative, so now we will calculate the antiderivative of secx tanx.
The process of finding antiderivative of sec(x) tan(x) is shown in steps below-
Step 1 : Find the derivative of sec(x), that is d(sec x) / dx = sec(x) tan(x) dx. Therefore g(x) = sec(x) tan(x).
Step 2 : Now to find the antiderivative of sec(x) tan(x) , we will have to find the integral of sec(x) tan(x). That is ∫sec(x) tan(x) dx = sec(x).
Proof of above operation-
=> Let g(x)= sec x tan x dx
=> g(x) = ∫ sin x / cos ² x dx ( sec x = 1 / cos x , tan x = sin x / cos x)
Let u = cos x
Therefore du = - sin x dx
=> g(x) = - ∫ 1 / u ² du
=> g(x) = - ∫ u-2 du
=> g(x) = 1 / u + C
=> g(x) = 1 / cos x + C
=> g(x) = sec x + C
Here we have used method of Integration by parts to calculate the antiderivative of sec x tan x.
Hence to find the antiderivative of secxtanx we have to calculate the integral of secxtanx which is sec (x) + C where as C is any arbitary constant generated after integration.