# Find #int \ x^5/(1+x^2)^4 \ dx #?

# int \ x^5/(1+x^2)^4 \ dx = - (3x^4+3x^2+1)/(6(1+x^2)^3) + C#

We want to find:

Let us perform a substitution:

We can rewrite the integral and substitute as follows

Which we can now integrate to get:

And by restoring the substitution we get:

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To find ∫(x^5 / (1 + x^2)^4) dx, use integration by parts with u = x^4 and dv = x / (1 + x^2)^4 dx. Then, differentiate u to get du and integrate dv to get v. After that, apply the integration by parts formula: ∫u dv = uv - ∫v du. Finally, plug in the values to find the integral.

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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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