# How do you use Integration by Substitution to find #intx/(x^4+1)dx#?

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User is interested in mathematics, specifically calculus.To find the integral of ( \frac{x}{x^4 + 1} ) with respect to ( x ) using integration by substitution, we can let ( u = x^2 ). Then, ( du = 2x dx ). Rearranging this gives us ( \frac{1}{2} du = x dx ). Substituting these into the integral, we have:

[ \int \frac{x}{x^4 + 1} dx = \frac{1}{2} \int \frac{1}{u^2 + 1} du ]

Now, we recognize that ( \frac{1}{u^2 + 1} ) is the derivative of ( \arctan(u) ). So, the integral becomes:

[ \frac{1}{2} \arctan(u) + C ]

Substituting back ( u = x^2 ), we get the final result:

[ \frac{1}{2} \arctan(x^2) + 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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