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Question: How do you test the improper integral \[\int{\dfrac{x}{\sqrt{1-{{x}^{2}}}}dx}\] from \[\left[ 0,1 \r...

How do you test the improper integral ∫x1−x2dx\int{\dfrac{x}{\sqrt{1-{{x}^{2}}}}dx} from [0,1]\left[ 0,1 \right] and evaluate if possible?

Explanation

Solution

In this problem, we can integrate the given improper integral using substitution. We can first take a trigonometric value for x and differentiate it for the value of dx, we can substitute in the given equation. we can then integrate the terms. We are given a limit value, we can apply it and subtract the limits after applying it to get the exact value of the given integral.

Complete step-by-step answer:
We know that the given integral is,
∫x1−x2dx\int{\dfrac{x}{\sqrt{1-{{x}^{2}}}}dx}….. (1)
We can now substitute for x,
Let x = sin⁡y\sin y,
Then dx=cos⁡ydydx=\cos ydy
Here we have to change the limit as we have substitution.
We know that the given limit is [0,1]\left[ 0,1 \right].
As for x = 0, then sin⁡0=0\sin 0=0
As for x = 1, then sin⁡1=π2\sin 1=\dfrac{\pi }{2}
Therefore, the new limit is [0,π2]\left[ 0,\dfrac{\pi }{2} \right].
We can now substitute the above values in (1), we get
=∫(sin⁡y1−sin⁡2y)cos⁡ydy= \int{\left( \dfrac{\sin y}{\sqrt{1-{{\sin }^{2}}y}} \right)\cos ydy} …… (2)
We can now write the denominator as,
1−sin⁡2y=cos⁡y\sqrt{1-{{\sin }^{2}}y}=\cos y
We can now substitute this value in (2), we get
=∫(sin⁡ycos⁡y)cos⁡ydy= \int{\left( \dfrac{\sin y}{\cos y} \right)\cos ydy}
We can now cancel the similar terms in the above step, we get
=∫sin⁡ydy= \int{\sin ydy}
We can now evaluate this using the new limits [0,π2]\left[ 0,\dfrac{\pi }{2} \right], we get
=∫0π2sin⁡ydy= \int\limits_{0}^{\dfrac{\pi }{2}}{\sin ydy}
We can now integrate the above step and apply the limits, we get
=[−cos⁡y]0π2= \left[ -\cos y \right]_{0}^{\dfrac{\pi }{2}}
We can now apply the limits, we get
=−cos⁡π2−(−cos⁡0)=1= -\cos \dfrac{\pi }{2}-\left( -\cos 0 \right)=1
Where, cos 0 =1.
Therefore, the improper integral ∫x1−x2dx\int{\dfrac{x}{\sqrt{1-{{x}^{2}}}}dx} from [0,1]\left[ 0,1 \right] is 1.

Note: We should always remember the differentiation and the integration formula to integrate or differentiate. We can solve the definite integral by applying the limit values, we can then subtract the upper limit from the lower limit to get the exact value of the given integral.