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The final value of the integral is $1$. This is obtained by evaluating the recognized sine series at the endpoints of the interval: $\sin(\pi/2) - \sin(0) = 1 - 0 = 1$.

Conditions: The integration interval is $[0,\pi/2]$.; The integrated series has been correctly identified as the Maclaurin series for $\sin x$.; The Fundamental Theorem of Calculus is applied to evaluate the definite integral.

The final value of the integral is $1$. This is obtained by evaluating the recognized sine series at the endpoints of the interval: $\sin(\pi/2) - \sin(0) = 1 - 0 = 1$.

Conditions: The integration interval is $[0,\pi/2]$.; The integrated series has been correctly identified as the Maclaurin series for $\sin x$.; The Fundamental Theorem of Calculus is applied to evaluate the definite integral.

The DCT justifies exchanging the definite integral over $[a,b]$ with the infinite summation, allowing term-by-term integration. Specifically, it establishes that $\int_a^b \sum f_n(x)\,dx = \sum \int_a^b f_n(x)\,dx$.

Conditions: The previously stated convergence, domination, and finite-integral conditions are assumed to hold.; The summation is over the sequence of functions $f_n$.; The integration is over the closed interval $[a,b]$.

The DCT justifies exchanging the definite integral over $[a,b]$ with the infinite summation, allowing term-by-term integration. Specifically, it establishes that $\int_a^b \sum f_n(x)\,dx = \sum \int_a^b f_n(x)\,dx$.

Conditions: The previously stated convergence, domination, and finite-integral conditions are assumed to hold.; The summation is over the sequence of functions $f_n$.; The integration is over the closed interval $[a,b]$.

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.

The integral is computed by substituting the Maclaurin series for $\cos x$, using the DCT to interchange the sum and integral, integrating each monomial term using the power rule, simplifying the factorial denominator, recognizing the resulting series as the sine series, and evaluating it at the endpoints $0$ and $\pi/2$.

Conditions: The integration interval is $[0,\pi/2]$.; The cosine power series is $\cos x = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} x^{2n}$.; The DCT is applicable because the series terms are dominated by $g(x)=1$ on $[0,\pi/2]$.

The integral is computed by substituting the Maclaurin series for $\cos x$, using the DCT to interchange the sum and integral, integrating each monomial term using the power rule, simplifying the factorial denominator, recognizing the resulting series as the sine series, and evaluating it at the endpoints $0$ and $\pi/2$.

Conditions: The integration interval is $[0,\pi/2]$.; The cosine power series is $\cos x = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} x^{2n}$.; The DCT is applicable because the series terms are dominated by $g(x)=1$ on $[0,\pi/2]$.

The sum and integral can be interchanged because the Dominated Convergence Theorem applies. The terms of the cosine power series are bounded in magnitude by $1$ on the interval $[0,\pi/2]$, making the constant function $g(x)=1$ a valid dominating function with a finite integral over the interval.

Conditions: The integration interval is $[0,\pi/2]$.; The series terms are $\frac{(-1)^n}{(2n)!}x^{2n}$.; The dominating function is $g(x)=1$.

The sum and integral can be interchanged because the Dominated Convergence Theorem applies. The terms of the cosine power series are bounded in magnitude by $1$ on the interval $[0,\pi/2]$, making the constant function $g(x)=1$ a valid dominating function with a finite integral over the interval.

Conditions: The integration interval is $[0,\pi/2]$.; The series terms are $\frac{(-1)^n}{(2n)!}x^{2n}$.; The dominating function is $g(x)=1$.