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The $(2n+1)!$ denominator comes from combining the original factorial $(2n)!$ from the cosine series coefficient with the new factor $2n+1$ introduced by the power rule of integration. Algebraically, $(2n)! \times (2n+1) = (2n+1)!$.

Conditions: The integration is performed on the term $x^{2n}$.; The original coefficient contains $(2n)!$ in the denominator.; The power rule is applied, raising the exponent to $2n+1$ and dividing by $2n+1$.

The $(2n+1)!$ denominator comes from combining the original factorial $(2n)!$ from the cosine series coefficient with the new factor $2n+1$ introduced by the power rule of integration. Algebraically, $(2n)! \times (2n+1) = (2n+1)!$.

Conditions: The integration is performed on the term $x^{2n}$.; The original coefficient contains $(2n)!$ in the denominator.; The power rule is applied, raising the exponent to $2n+1$ and dividing by $2n+1$.

The integrated cosine series becomes the sine series because integrating the even-power terms $x^{2n}$ of the cosine series produces odd-power terms $x^{2n+1}$. Combined with the factorial simplification $(2n)!(2n+1) = (2n+1)!$, the resulting series exactly matches the standard Maclaurin series for $\sin x$.

Conditions: The starting series is the Maclaurin series for $\cos x$.; Term-by-term integration is justified by the DCT.; The factorial identity $(2n)!(2n+1) = (2n+1)!$ is applied.

The integrated cosine series becomes the sine series because integrating the even-power terms $x^{2n}$ of the cosine series produces odd-power terms $x^{2n+1}$. Combined with the factorial simplification $(2n)!(2n+1) = (2n+1)!$, the resulting series exactly matches the standard Maclaurin series for $\sin x$.

Conditions: The starting series is the Maclaurin series for $\cos x$.; Term-by-term integration is justified by the DCT.; The factorial identity $(2n)!(2n+1) = (2n+1)!$ is applied.