What is wrong inthe given equation.
The given expression for ∫π/ 3πsecθtanθdθ is incorrect, since ∫π/ 3πsecθtanθdθ does not exist as a difference of anti-derivative.
Use the fundamental theorem of calculus part 2 to explain the wrong in the given equation.
2) Fundamental theorem of Calculus, Part 2
If f is continuous on a,b, then
where F is any antiderivative of f, that is, a function F such that F'=f
∫π/ 3πsecθtanθdθ=secθ]ππ/ 3=-3
The graph of function fθ=secθtanθ, π/3≤θ≤π is given by,
By trigonometric formula,
Therefore, the function fθ=secθtanθ is not continuous on π/3,π
So, the Fundamental theorem of Calculus, Part 2 cannot be applied. So we cannot express the given integral as
∫π/ 3πsecθtanθdθ=Fπ-Fπ/3 where F is antiderivative of f, that is, a function F such that F'(θ)=f(θ). Besides the definite integral doesn’t exists.
∫π/ 3πsecθtanθdθ does not exist as a difference of anti-derivative, therefore the given equation is wrong.