Evaluate the integral? : # int 1/(xsqrt(1-x^4)) dx#

Answer 1

# int \ 1/(xsqrt(1-x^4)) \ dx = 1/4 (ln|sqrt(1-x^4)-1| -ln|sqrt(1-x^4)+1 )+ C#

We seek:

# I = 1/(xsqrt(1-x^4)) \ dx #

Let us attempt an substitution of the form:

# u = sqrt(1-x^4) => u^2 = 1-x^4 #, and #x^4 = 1-u^2 #
And differentiating wrt #x# we get:
# 2u(du)/dx = -4x^3 => (2u)/(-4x^4) (du)/dx = (-4x^3)/(-4x^4) # # :. -(u)/(2x^4) (du)/dx = 1/x #

If we substitute this into the integral, we get:

# I = int \ (1/u) ( -(u)/(2(1-u^2))) \ du # # \ \ = 1/2 \ int \ 1/(u^2-1) \ du #

Now, we can decompose this new integrand 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)) #

Leadin to:

# 1 = (u-1) + B(u+1) #
Where #A,B# are constants that are to be determined. We can find them by substitutions (In practice we do this via the "cover up" method:
Put # u = +1 => 0 = 2B => B=1/2 # Put # u = -1 => 0 = -2A => A=-1/2 #

Thus:

# I = 1/2 \ int \ 1/(u^2-1) \ du # # \ \ = 1/2 \ int \ {(-1/2)/(u+1) + (1/2)/(u-1)} \ du # # \ \ = 1/4 \ int \ {1/(u-1) -1/(u+1)} \ du #

Which we can directly integrate:

# I = 1/4 (ln|u-1| -0 ln|u+1 )+ C#

Restoring the substitution:

# I = 1/4 (ln|sqrt(1-x^4)-1| -ln|sqrt(1-x^4)+1 )+ C#
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Answer 2

To evaluate the integral ( \int \frac{1}{x \sqrt{1 - x^4}} , dx ), use the substitution ( u = x^2 ). Then, ( du = 2x , dx ). Rewrite the integral as ( \frac{1}{2} \int \frac{1}{\sqrt{u(1-u^2)}} , du ). This becomes ( \frac{1}{2} \int \frac{1}{\sqrt{u} \sqrt{1-u^2}} , du ). Apply a trigonometric substitution ( u = \sin(\theta) ) to get ( du = \cos(\theta) , d\theta ). The integral transforms into ( \frac{1}{2} \int \frac{1}{\cos(\theta)} , d\theta ). This simplifies to ( \frac{1}{2} \int \sec(\theta) , d\theta ). Integrate ( \sec(\theta) ) to get ( \ln|\sec(\theta) + \tan(\theta)| + C ). Replace ( \theta ) with ( \arcsin(u) ). Finally, substitute ( u = x^2 ) and simplify the result.

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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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