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受欢迎的微积分 >

integral of cos^4(2x)

解答

\int cos^{4} (2 x) d x

解答

\frac{1}{2} (cos^{3} (2 x) sin (2 x) + \frac{3}{8} (2 x - \frac{1}{4} sin (8 x))) + C

求解步骤

\int cos^{4} (2 x) d x

使用换元积分法

= \int cos^{4} (u) \frac{1}{2} d u

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int cos^{4} (u) d u

化简

cos^{4} (u) : cos^{3} (u) cos (u)

= \frac{1}{2} \cdot \int cos^{3} (u) cos (u) d u

使用分布积分法

= \frac{1}{2} (cos^{3} (u) sin (u) - \int - 3 cos^{2} (u) sin^{2} (u) d u)

\int - 3 cos^{2} (u) sin^{2} (u) d u = - \frac{3}{8} (u - \frac{1}{4} sin (4 u))

= \frac{1}{2} (cos^{3} (u) sin (u) - (- \frac{3}{8} (u - \frac{1}{4} sin (4 u))))

u = 2 x

代回

= \frac{1}{2} (cos^{3} (2 x) sin (2 x) - (- \frac{3}{8} (2 x - \frac{1}{4} sin (4 \cdot 2 x))))

化简

= \frac{1}{2} (cos^{3} (2 x) sin (2 x) + \frac{3}{8} (2 x - \frac{1}{4} sin (8 x)))

解答补常数

= \frac{1}{2} (cos^{3} (2 x) sin (2 x) + \frac{3}{8} (2 x - \frac{1}{4} sin (8 x))) + C

作图

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