\int_{0}^{\frac{\Pi }{2}}\frac{dx}{1+cos^{2}x}
In arihant, he divides Nr and Dr by cos2x which is absurd as it would be like dividing by 0 as cos2(90) =0 (90 is in the domain of x)
Secondly, after this step, the integrand is improper as it is not defined at x=90
But still he uses substitution and finds out the answer.
Is it that definition of function in integrand at the upper and lower limits doesnot matter? Do we have to use limits for a correct approach?
Expert opinion needed.
THanks
106yes it is correct.
\int_{0}^{\frac{\pi}{2}}{\frac{dx}{1+cos^2x}} \equiv \int_{0}^{\frac{\pi}{2}_{-}}{\frac{dx}{1+cos^2x}}
a single point of discontinuity/being infinite or not being defined doesnt affect the value of the integral. and hence it is perfectly fine to divide by cos^2x
1Yes it doesnot affect the value of the integral. But the method can be proved wrong mathematically ( method followed after dividing by cos2x )
According to Fundamental Theorem of Calculus, the integrand must be continuous on [a, b], which is what you do when you find an antiderivative for f and evaluate it at the two endpoints. Otherwise you are dealing with an improper integral. If f is not defined at the endpoints (or at points inside the interval), you must use limits to evaluate the integral which the author doesn't mention/use.