What is integral (0 to 1) x^3/(x^4+1) ?  

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You should come up with the following substitution, such that:

`x^4 + 1 = t`

Differentiating both sides yields:

`4x^3 dx = dt => x^3 dx = (dt)/4`

Replacing the limits of integration yields:

`x = 0 => t = 1`

`x = 1 => t = 1^4 + 1 = 2`

Replacing the variable to integrand, yields:

`int_1^2 (dt)/(4t) = (1/4)int_1^2 (dt)/t`

`(1/4)int_1^2 (dt)/t = (1/4)ln t|_1^2`

Using the fundamental theorem of calculus, yields:

`(1/4)int_1^2 (dt)/t = (1/4)(ln 2 - ln 1)`

Since `ln 1 = 0` yields:

`(1/4)int_1^2 (dt)/t = (1/4)(ln 2) => (1/4)int_1^2 (dt)/t = ln 2^(1/4)`

`(1/4)int_1^2 (dt)/t = ln root(4)2`

Hence, evaluating the given definite integral, yields` int_0^1 x^3/(x^4 + 1)dx = ln root(4)2.`

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