We have to find the Antiderivative of secx tanx. To find the antiderivative of secx tanx we must first understand the meaning or definition of antiderivative.
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.