# How do you integrate #int x(1-x^2)^(1/4)dx#?

A substitution will do.

We then have

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To integrate ( \int x(1-x^2)^{\frac{1}{4}} , dx ), use a substitution method.

Let ( u = 1 - x^2 ).

Then, ( du = -2x , dx ).

Solve for ( dx ): [ dx = -\frac{du}{2x} ]

Replace ( x ) and ( dx ) in the integral:

[ \int x(1-x^2)^{\frac{1}{4}} , dx = \int -\frac{1}{2} u^{\frac{1}{4}} , du ]

Now, integrate with respect to ( u ):

[ \int -\frac{1}{2} u^{\frac{1}{4}} , du = -\frac{1}{2} \cdot \frac{u^{\frac{5}{4}}}{\frac{5}{4}} + C ]

Replace ( u ) with ( 1 - x^2 ):

[ = -\frac{1}{2} \cdot \frac{(1 - x^2)^{\frac{5}{4}}}{\frac{5}{4}} + C ]

Simplify the expression:

[ = -\frac{2}{5} (1 - x^2)^{\frac{5}{4}} + C ]

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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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