Quiz(73/100)[tex] \: [/tex]Soal terlampirr .... Question from @Aikey

May 2022 1 10 Report

Quiz(73/100)
[tex] \: [/tex]
Soal terlampirr

Hasil dari $\displaystyle{\int\limits^{\frac{\pi}{6}}_0 {sin\left ( x+\frac{\pi}{3} \right )cos\left ( x+\frac{\pi}{3} \right )} \, dx }$ adalah $\displaystyle{\boldsymbol{C.~\frac{1}{8}} }$ .

PEMBAHASAN

Integral merupakan operasi yang menjadi kebalikan dari operasi turunan/diferensial. Sehingga integral sering juga disebut sebagai antiturunan.

$\displaystyle{f(x)=\int\limits {\left [ \frac{df(x)}{dx} \right ]} \, dx}$

Sifat - sifat operasi pada integral adalah sebagai berikut :

$(i)~\displaystyle{\int\limits {ax^n} \, dx=\frac{a}{n+1}x^{n+1}+C},~~~dengan~C=konstanta$

$(ii)~\displaystyle{\int\limits {kf(x)} \, dx=k\int\limits {f(x)} \, dx}$

$(iii)~\displaystyle{\int\limits {\left [ f(x)\pm g(x) \right ]} \, dx=\int\limits {f(x)} \, dx\pm\int\limits {g(x)} \, dx}$

$(iv)~\displaystyle{\int\limits^b_a {f(x)} \, dx=F(b)-F(a)}$

DIKETAHUI

$\displaystyle{\int\limits^{\frac{\pi}{6}}_0 {sin\left ( x+\frac{\pi}{3} \right )cos\left ( x+\frac{\pi}{3} \right )} \, dx= }$

DITANYA

Tentukan hasilnya.

PENYELESAIAN

Gunakan identitas $sin2\theta=2sin\theta cos\theta$ .

$\displaystyle{\int\limits^{\frac{\pi}{6}}_0 {sin\left ( x+\frac{\pi}{3} \right )cos\left ( x+\frac{\pi}{3} \right )} \, dx }$

$\displaystyle{=\int\limits^{\frac{\pi}{6}}_0 {\frac{1}{2}sin\left [ 2\left ( x+\frac{\pi}{3} \right ) \right ]} \, dx }$

$\displaystyle{=\frac{1}{2}\int\limits^{\frac{\pi}{6}}_0 {sin\left ( 2x+\frac{2\pi}{3} \right ) } \, dx }$

$----------------$

Misal :

$\displaystyle{u=2x+\frac{2\pi}{3}~\to~du=2dx }$

$\displaystyle{untuk~x=0~\to~u=2(0)+\frac{2\pi}{3}=\frac{2\pi}{3} }$

$\displaystyle{untuk~x=\frac{\pi}{6}~\to~u=2\left ( \frac{\pi}{6} \right )+\frac{2\pi}{3}=\pi }$

$----------------$

$\displaystyle{=\frac{1}{2}\int\limits^{\pi}_{\frac{2\pi}{3}} {sinu } \, \left ( \frac{du}{2} \right ) }$

$\displaystyle{=\frac{1}{4}\int\limits^{\pi}_{\frac{2\pi}{3}} {sinu } \, du }$

$\displaystyle{=-\frac{1}{4}cosu\Bigr|^{\pi}_{\frac{2\pi}{3}} }$

$\displaystyle{=-\frac{1}{4}\left [ cos\left ( \pi \right )-cos\left ( \frac{2\pi}{3} \right ) \right ] }$

$\displaystyle{=-\frac{1}{4}\left ( -1+\frac{1}{2} \right )}$

$\displaystyle{=-\frac{1}{4}\left ( -\frac{1}{2} \right )}$

$\displaystyle{=\frac{1}{8}}$

KESIMPULAN

Hasil dari $\displaystyle{\int\limits^{\frac{\pi}{6}}_0 {sin\left ( x+\frac{\pi}{3} \right )cos\left ( x+\frac{\pi}{3} \right )} \, dx }$ adalah $\displaystyle{\boldsymbol{C.~\frac{1}{8}} }$ .