Question

вычислить интеграл:

∫от π/8 до π/4 cos4xdx 

Answer 1

= 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

Answer 2

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