# How do you find the antiderivative of #int x(x^2+1)^100 dx#?

Whatever you do, don't try to expand the bracket by the binomial expansion!

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To find the antiderivative of ( \int x(x^2+1)^{100} , dx ), you can use the substitution method. Let ( u = x^2 + 1 ). Then, ( du = 2x , dx ). Rearrange this to express ( dx ) in terms of ( du ), giving ( dx = \frac{1}{2x} , du ). Substitute these expressions into the integral:

[ \int x(x^2+1)^{100} , dx = \int x u^{100} \cdot \frac{1}{2x} , du ]

Simplify:

[ \frac{1}{2} \int u^{100} , du ]

Now integrate ( u^{100} ) with respect to ( u ):

[ \frac{1}{2} \cdot \frac{u^{101}}{101} + C ]

Substitute back ( u = x^2 + 1 ):

[ \frac{1}{202} (x^2 + 1)^{101} + C ]

So, the antiderivative of ( \int x(x^2+1)^{100} , dx ) is ( \frac{1}{202} (x^2 + 1)^{101} + C ), where ( C ) is the constant of integration.

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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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