int(4y^(4)-y^(2))/(4y^(2)+1)dy

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Split Integrand: Simplify the integrand if possible. The integrand (4y^(4)-y^(2))/(4y^(2)+1) can be split into two separate fractions: 4y^(4)/(4y^(2)+1) - y^(2)/(4y^(2)+1).Integrate First Term: Integrate the first term 4y^(4)/(4y^(2)+1). Notice that the numerator is the derivative of the denominator. Therefore, the integral of 4y^(4)/(4y^(2)+1) with respect to y is simply the natural logarithm of the absolute value of the denominator. ∫4y^(4)/(4y^(2)+1) dy = ln|4y^(2)+1| + C_1, where C_1 is an arbitrary constant.Integrate Second Term: Integrate the second term -y^(2)/(4y^(2)+1). This term does not simplify directly, and it does not match a standard integral form. We can try a substitution or partial fractions, but in this case, partial fractions do not apply since the degree of the numerator is less than the degree of the denominator. We will attempt a substitution. Let u = 4y^(2)+1, then du = 8y dy. We need to express y^(2) dy in terms of u and du. Since u = 4y^(2)+1, y^(2) = (u-1)/4. Also, dy = du/(8y).Substitute in Integral: Substitute y^(2) and dy in terms of u and du in the integral. We have y^(2) = (u-1)/4 and dy = du/(8y), so we can write: ∫-y^(2)/(4y^(2)+1) dy = ∫-(u-1)/4 * 1/u * du/(8y). However, we have a y in the denominator of du/(8y) which we have not expressed in terms of u. This is a mistake because we cannot integrate with respect to u while still having a variable y in the expression.

$\int \frac{4y^{4}-y^{2}}{4y^{2}+1}\,dy$

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$\int \frac{4y^{4}-y^{2}}{4y^{2}+1}\,dy$