Question

Bab Integral
SMA kelas 11

Hasil dari [tex]\int\limits {(Sin2x-cosx)} \, dx[/tex] adalah ...
a. [tex]-\frac{1}{2} Cos2x-sinx+c[/tex]
b. -2 Cos 2x - sin x + c
c. [tex]\frac{1}{2} cos2x+sinx+c[/tex]
d. [tex]\frac{1}{2} cos2x-sinx+c[/tex]

Answer 1

[tex]\int(\sin2x-\cos x)\,dx=\displaystyle\bf{-}\frac{1}{2}\cos 2x-\sin x+C[/tex](opsi a) PembahasanIntegral Fungsi Trigonometri [tex]\begin{aligned}&\int(\sin2x-\cos x)\,dx\\&{=\ }\int\sin2x\,dx-\int\cos x\,dx\\&\quad\left[\ \begin{aligned}&{\sf Ambil}\ u=2x\\&{\Rightarrow\ }du=2dx\\&{\Rightarrow\ }dx=\frac{1}{2}du\\&{\Rightarrow\ }\int\sin2x\,dx\\&{\quad}=\int\sin u\cdot\frac{1}{2}du\\&{\quad}=\frac{1}{2}\int\sin u\,du\\&{\quad}=\frac{1}{2}(-\cos u)\\&{\quad}=-\frac{1}{2}\cos u\\&{\quad}=-\frac{1}{2}\cos 2x\\\end{aligned}\right.\\&{=\ }-\frac{1}{2}\cos 2x-\int\cos x\,dx\end{aligned}[/tex][tex]\begin{aligned}&{=\ }\boxed{\ \bf{-}\frac{1}{2}\cos 2x-\sin x+C\ }\end{aligned}[/tex]