How do you evaluate the integral #int 1/(xsqrt(1-x^2))#?

Answer 1

The answer is# =-1/2ln(sqrt(1-x^2)+1)+1/2ln(|sqrt(1-x^2)-1|)+C#

We perform this integral by substitution

Let #u=sqrt(1-x^2)#
#u^2=1-x^2#
#du=(-2xdx)/(2sqrt(1-x^2))=(-xdx)/(sqrt(1-x^2))#

Therefore,

#intdx/(xsqrt(1-x^2))=intsqrt(1-x^2)(du)/(-x^2sqrt(1-x^2))#
#=int-(du)/x^2#
#=int(du)/(u^2-1)#

Now, we perform a de composition into partial fractions

#1/(u^2-1)=1/((u+1)(u-1))=A/(u+1)+B/(u-1)#
#=(A(u-1)+B(u+1))/((u+1)(u-1))#

The denominators are the same, we can compare the numerators

#1=A(u-1)+B(u+1)#
Let #u=-1#, #=>#, #1=-2A#, #=>#, #A=-1/2#
Let #u=1#, #=>#, #1=2B#, #=>#, #B=1/2#

Therefore,

#1/(u^2-1)=(-1/2)/(u+1)+(1/2)/(u-1)#

So,

#int(du)/(u^2-1)=-1/2int(du)/(u+1)+1/2int(du)/(u-1)#
#=-1/2ln(u+1)+1/2ln(u-1)#
#=-1/2ln(sqrt(1-x^2)+1)+1/2ln(|sqrt(1-x^2)-1|)+C#
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Answer 2

To evaluate the integral ( \int \frac{1}{x\sqrt{1-x^2}} ), we can use trigonometric substitution. Let ( x = \sin(\theta) ), then ( dx = \cos(\theta) d\theta ) and ( \sqrt{1 - x^2} = \sqrt{1 - \sin^2(\theta)} = \sqrt{\cos^2(\theta)} = \cos(\theta) ).

After substitution, the integral becomes:

[ \int \frac{1}{\sin(\theta) \cdot \cos(\theta)} \cdot \cos(\theta) , d\theta ]

[ = \int \frac{1}{\sin(\theta)} , d\theta ]

[ = \int \csc(\theta) , d\theta ]

Now, this is a standard integral which integrates to (-\ln|\csc(\theta) + \cot(\theta)| + C). However, since we are integrating with respect to (x), we need to express the result in terms of (x).

Recall that (x = \sin(\theta)). Therefore, ( \csc(\theta) = \frac{1}{\sin(\theta)} = \frac{1}{x} ). Also, ( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} = \frac{\sqrt{1 - x^2}}{x} ).

Thus, the integral becomes:

[ = -\ln|\frac{1}{x} + \frac{\sqrt{1 - x^2}}{x}| + C ]

[ = -\ln|\frac{\sqrt{1 - x^2} + 1}{x}| + C ]

This is the result after evaluating the integral ( \int \frac{1}{x\sqrt{1-x^2}} ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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