Antiderivatives: An Introduction to Indefinite Integrals

The rate of change of a function at a particular value x is known as the Derivative of that function. Anti-derivatives as the name suggests is the opposite of derivatives. An Anti derivative is commonly referred to as an Indefinite Integral. We can define an indefinite integral of F as follows: Any given function G is an indefinite integral of F or an indefinite integral of a function g if the derivative of that function G’ equals g. The notation of an indefinite integral of F or the indefinite integral of a function is, G(x) = Integral g(x) dx. From this notation, we can conclude that G(x) equals integral[g(x)]dx  if and only if G’(x) = g(x)

From the above we can understand that an Antiderivative is basically an Integral of a given function, which is set into a formula which helps us to take the indefinite integral of F. So, when we say the Integral it means indefinite integral of F. Here, we have to remember to add a constant “c” as every integral has an unknown constant which is added to the equation. Let us consider an example for a better understanding, given function y = x^2 + 3x + 5. The derivative y’ would be 2x +3. Let us now find the antiderivative of y’, that gives us integral[2x +3] dx = 2. X^(1+1)/(1+1) + 3. x^(0+1)/(0+1) + c = 2x^/2 + 3x/1 + c = x^2+3x +c, this function is same as the original function except that the constant 5 is missing, this is the reason why we need to add the constant “c” to the Integral of a function.  From this we can conclude that Anti Derivative is the reverse derivative or the indefinite integral.

When we solve an Integral, we eliminate the integral sign and dx to arrive to a function G(x), this function is the antiderivative.  For instance, indefinite integral of F of the function x^3 is given by x raised to the power (3+1) whole divided by (3+1), that is, integral(x^3)dx = x^(3+1)/(3+1)= x^4/4. In general we can write the indefinite integral of F of x^n  as x^(n+1)/(n+1)
Antiderivative of Sec X
indefinite integral of F is the Integral of Sec X
Multiplying sec(x) with 1 which is [sec(x)+tan(x)]/[tan(x)+sec(x)]
Integral[sec(x)] dx = Integral[Sec x][sec(x) +tan(x)]/[tan(x)+sec(x)]      

Let u= sec(x) + tan(x)
Differentiating on both sides,
du = [sec(x)tan(x) + sec^2(x)]dx
Substituting u= [sec(x) + tan(x)]du = [sec(x)tan(x) + sec^2(x)]dx ,
Integral[sec(x)][sec(x)+tan(x)]/[tan(x)+sec(x)] dx
 = Integral[sec^2(x)+sec(x)tan(x)]dx/[sec(x)+tan(x)]
= Integral[du/u]
Solving the integral we get,
 = ln|u|+ c
Again substituting u= sec(x) + tan(x)
= ln|sec(x)+tan(x)| + c
So, the Antiderivative of Sec X is the Integral[sec x]dx = ln|sec(x)+tan(x)| + c