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受欢迎的微积分 >

integral of 128cos^4(8x)

解答

\int 128 cos^{4} (8 x) d x

解答

16 (cos^{3} (8 x) sin (8 x) + 3 x - \frac{3}{32} sin (32 x)) + C

求解步骤

\int 128 cos^{4} (8 x) d x

提出常数:

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

= 128 \cdot \int cos^{4} (8 x) d x

使用换元积分法

= 128 \cdot \int cos^{4} (u) \frac{1}{8} d u

提出常数:

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

= 128 \cdot \frac{1}{8} \cdot \int cos^{4} (u) d u

化简

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

= 128 \cdot \frac{1}{8} \cdot \int cos^{3} (u) cos (u) d u

使用分布积分法

= 128 \cdot \frac{1}{8} (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))

= 128 \cdot \frac{1}{8} (cos^{3} (u) sin (u) - (- \frac{3}{8} (u - \frac{1}{4} sin (4 u))))

u = 8 x

代回

= 128 \cdot \frac{1}{8} (cos^{3} (8 x) sin (8 x) - (- \frac{3}{8} (8 x - \frac{1}{4} sin (4 \cdot 8 x))))

化简

128 \cdot \frac{1}{8} (cos^{3} (8 x) sin (8 x) - (- \frac{3}{8} (8 x - \frac{1}{4} sin (4 \cdot 8 x)))) : 16 (cos^{3} (8 x) sin (8 x) + 3 x - \frac{3}{32} sin (32 x))

= 16 (cos^{3} (8 x) sin (8 x) + 3 x - \frac{3}{32} sin (32 x))

解答补常数

= 16 (cos^{3} (8 x) sin (8 x) + 3 x - \frac{3}{32} sin (32 x)) + C

作图

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