Why is it very easy to conclude that \int_{1}^{\propto}x^{2}dx d

hroncits8y

hroncits8y

Answered question

2021-11-18

Why is it very easy to conclude that 1x2dx diverges without making any integration calculations?

Answer & Explanation

Phisecome

Phisecome

Beginner2021-11-19Added 18 answers

Step 1
This question is taken from the calculus and sub-topic is improper integral in which we have to give the reason that given integral is diverges in nature without making any integration calculation. We can write the integral below
consider
u=1x2dx
now to solve we move to next step-2
Step 2
the given integral is improper integral in which we directly say about the integral is converge or diverges by simply checking the limit of the integral function at the upper limit of the integral so first we write it as.
f(x)=x2
u=1f(x)dx
now if the limit of the function is finite then we can found the integral value so it converges and if the limits become infinite or the limit does not exist of the function then we can not find the value of the integral and it diverges.
so first we have to check the limit of the function that exists or not so
u=1f(x)dx
u=1x2dx
f(x)=x2
now taking the limit of the function consider k is limiting value then
k=limx→∝f(x)
put the f(x)=x2
k=limx→∝f(x)
k=limx→∝x2
now put the limits
k=2=∝
k=∝
or
k=limx→∝x2=∝
now the value of the kis infinite it means that we can not find the value of the given integral so the integral is diverges.
now we move to the next step for the result
Step 3 Result
Result: from the above analysis we concluded that the given integral is diverges since its functional limiting value is infinite so itis very easy to say about the given integral is diverges.

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