вычислить интеграл: ∫от π/8 до π/4 cos4xdx . Created by usmile. algebra-ru

4 cos4xdx ...

nuli977

= sin4x/4 |от пи/8 до пи/4 = 1/4 * (sin4*pi/4 - sin 4*pi/8) = 1/4 *(sinpi-sin(pi/2)) = 1/4*(0-1) = -1/4

4 votes Thanks 3

konrad509

$\\\int \limits_{\frac{\pi}{8}}^{\frac{\pi}{4}} {\cos 4x}\, dx=(*)\\ t=4x\\ dt=4\, dx\\ \int \limits_{\frac{\pi}{8}}^{\frac{\pi}{4}} \frac{1}{4}{\cos t}\, dt=\\ \frac{1}{4}\int \limits_{\frac{\pi}{8}}^{\frac{\pi}{4}} {\cos t}\, dt=\\ \frac{1}{4}\Big[\sin t\Big]_{\frac{\pi}{8}}^{\frac{\pi}{4}}=\\ (*)=\frac{1}{4}\Big[\sin 4x\Big]_{\frac{\pi}{8}}^{\frac{\pi}{4}}=\\ \frac{1}{4}(\sin (4\cdot\frac{\pi}{4})-\sin (4\cdot\frac{\pi}{8}))=\\ \frac{1}{4}(\sin \pi-\sin \frac{\pi}{2})=\\ \frac{1}{4}(0-1)=\\ \frac{1}{4}\cdot(-1)=\\ -\frac{1}{4}$

5 votes Thanks 7

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