1. The problem statement, all variables and given/known data
Hello , so in spivak's calculus we only learn to calculate the integral of x^2 and x in the first chapter , i was wonderin gwhat would the integral of ##x^(1/2)## ##\int_a^b##x^(1/2)## ~dx## , well i used the inverse function and a little bit of work i calculated from 0 to 2 using the integral of x^2 on 0 to 2^(1/2) and integral of 2^(1/2) on 0 to 2 turned out it's the correct answer using wolfram alpha , i checked other variables same , so i tried to generalize this , i don't know if this is correct but here it is (f-1 = f^-1 inverse function of f , it wouldn't let me write it on latex)
##\int_a^b f(x)~dx##=-##\int_c^d f^-1(x)~dx##+##\int_a^b f(b)-f(a)~dx##
where c = f^-1(a) and d=f^-1(b)
i didn't check for a lot of functions , but i was wondering if this is right ; thanks everyone :).
Hello , so in spivak's calculus we only learn to calculate the integral of x^2 and x in the first chapter , i was wonderin gwhat would the integral of ##x^(1/2)## ##\int_a^b##x^(1/2)## ~dx## , well i used the inverse function and a little bit of work i calculated from 0 to 2 using the integral of x^2 on 0 to 2^(1/2) and integral of 2^(1/2) on 0 to 2 turned out it's the correct answer using wolfram alpha , i checked other variables same , so i tried to generalize this , i don't know if this is correct but here it is (f-1 = f^-1 inverse function of f , it wouldn't let me write it on latex)
##\int_a^b f(x)~dx##=-##\int_c^d f^-1(x)~dx##+##\int_a^b f(b)-f(a)~dx##
where c = f^-1(a) and d=f^-1(b)
i didn't check for a lot of functions , but i was wondering if this is right ; thanks everyone :).
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