Materi : Integral Tentu
Level : Sulit
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Jawaban:
[tex]\large\text{$\begin{aligned}\int_{-\pi }^{\pi }\left(\sin x+\cos x\right)dx=\bf0\end{aligned}$}[/tex]
Pembahasan
Integral Tentu
[tex]\large\text{$\begin{aligned}\int_{-\pi}^{\pi}\left(\sin x+\cos x\right)dx=\int_{-\pi}^{\pi}\sin x\,dx+\int_{-\pi}^{\pi}\cos x\,dx\end{aligned}$}[/tex]
f(x) = sin x adalah fungsi ganjil, karena f(–x) = –f(x) untuk semua x ∈ ℝ.
Jika f(x) adalah fungsi ganjil dan kontinu pada interval –α ≤ x ≤ α, maka [tex]\displaystyle\int_{-\alpha}^{\alpha}{f(x)\,dx}=0[/tex] .
Oleh karena itu, [tex]\begin{aligned}\int_{-\pi}^{\pi}\sin x\,dx=0\end{aligned}[/tex] .
[tex]\begin{aligned}\int_{-\pi}^{\pi}\cos x\,dx&=\big[\sin x\big]_{-\pi}^{\pi}\\&=\lim_{x\to\pi}\sin x-\lim_{x\to-\pi}\sin x\\&=\sin(\pi)-\sin(-\pi)\\&=\sin(\pi)-\left(-\sin(\pi)\right)\\&=0-0\\&=0\end{aligned}[/tex]
Dengan demikian,
[tex]\large\text{$\begin{aligned}\int_{-\pi}^{\pi}\left(\sin x+\cos x\right)dx&=0+0\\&=\bf0\end{aligned}$}[/tex]
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