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

Question

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

Aurora Alvarado · Accepted Answer

# 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#

Christopher Agresta · Answer

undefined 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.