y = cos ( x ) y = sin ( 2 x ) x = 0 x = π 2 {\displaystyle {\begin{aligned}&\color {red}\mathbf {y=\cos(x)} &\color {royalblue}\mathbf {y=\sin(2x)} \\&x=0&x={\frac {\pi }{2}}\\\end{aligned}}}
cos ( x ) = sin ( 2 x ) x = π 2 x = π 6 {\displaystyle {\begin{aligned}\cos(x)&=\sin(2x)\\x&={\frac {\pi }{2}}\\x&={\frac {\pi }{6}}\\\end{aligned}}}
∫ 0 π 6 ( cos ( x ) − sin ( 2 x ) ) d x + ∫ π 6 π 2 ( sin ( 2 x ) − cos ( x ) ) d x {\displaystyle \int _{0}^{\frac {\pi }{6}}\left(\cos(x)-\sin(2x)\right)dx+\int _{\frac {\pi }{6}}^{\frac {\pi }{2}}\left(\sin(2x)-\cos(x)\right)dx}
∫ 0 π 6 ( cos ( x ) − sin ( 2 x ) ) d x = [ sin ( x ) + 1 2 cos ( 2 x ) ] | 0 π 6 = [ sin ( π 6 ) + 1 2 cos ( 2 π 6 ) ] − [ sin ( 0 ) + 1 2 cos ( 2 ( 0 ) ) ] = [ 1 2 + 1 2 ( 1 2 ) ] − [ 0 − 1 2 ( 1 ) ] = 1 2 + 1 4 − [ 0 + 1 2 ] = 1 4 {\displaystyle {\begin{aligned}\int _{0}^{\frac {\pi }{6}}\left(\cos(x)-\sin(2x)\right)dx&=\left[\sin(x)+{\frac {1}{2}}\cos(2x)\right]{\Bigg |}_{0}^{\frac {\pi }{6}}\\[2ex]&=\left[\sin({\frac {\pi }{6}})+{\frac {1}{2}}\cos({\frac {2\pi }{6}})\right]-\left[\sin(0)+{\frac {1}{2}}\cos(2(0))\right]\\[2ex]&=\left[{\frac {1}{2}}+{\frac {1}{2}}\left({\frac {1}{2}}\right)\right]-\left[0-{\frac {1}{2}}(1)\right]\\[2ex]&={\frac {1}{2}}+{\frac {1}{4}}-\left[0+{\frac {1}{2}}\right]\\&={\frac {1}{4}}\end{aligned}}}
/ b e g i n a l i g n ∫ / e n d a l i g n {\displaystyle /begin{align}\int _{}^{}/end{align}}
![{\displaystyle {\begin{aligned}\int _{0}^{\frac {\pi }{6}}\left(\cos(x)-\sin(2x)\right)dx&=\left[\sin(x)+{\frac {1}{2}}\cos(2x)\right]{\Bigg |}_{0}^{\frac {\pi }{6}}\\[2ex]&=\left[\sin({\frac {\pi }{6}})+{\frac {1}{2}}\cos({\frac {2\pi }{6}})\right]-\left[\sin(0)+{\frac {1}{2}}\cos(2(0))\right]\\[2ex]&=\left[{\frac {1}{2}}+{\frac {1}{2}}\left({\frac {1}{2}}\right)\right]-\left[0-{\frac {1}{2}}(1)\right]\\[2ex]&={\frac {1}{2}}+{\frac {1}{4}}-\left[0+{\frac {1}{2}}\right]\\&={\frac {1}{4}}\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/7f4f144d768f4b580ab565d962ee1adc273c220a)
