# How do you integrate #int xsqrt(1-x^4)# by trigonometric substitution?

We have:

Solving both of these integrals:

Write the cosine function as sine, since we are working with a sine-based substitution:

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To integrate ( \int x\sqrt{1-x^4} ) by trigonometric substitution, let ( x = \sin(\theta) ). Then, ( dx = \cos(\theta) d\theta ). Substituting these into the integral, you get ( \int \sin(\theta)\sqrt{1-\sin^4(\theta)} \cos(\theta) d\theta ). Simplify using trigonometric identities to express ( \sqrt{1-\sin^4(\theta)} ) in terms of ( \cos(\theta) ). Then, make further substitutions to simplify the integral into a form that can be integrated easily. Finally, integrate with respect to ( \theta ) and back-substitute to express the result in terms of ( x ).

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