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受欢迎的微积分 >

integral of 5sin^3(5xco)s^35x

解答

\int 5 sin^{3} (5 x co) s^{3} 5 x d x

解答

s^{3} \frac{1}{c ^{2} o ^{2}} (- 5 co x cos (5 co x) + \frac{5}{3} co x cos^{3} (5 co x) + \frac{2}{3} sin (5 co x) + \frac{1}{9} sin^{3} (5 co x)) + C

求解步骤

\int 5 sin^{3} (5 x co) s^{3} \cdot 5 x d x

提出常数:

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

= 5 \cdot 5 s^{3} \cdot \int sin^{3} (5 co x) x d x

化简

= 25 s^{3} \cdot \int sin^{3} (5 co x) x d x

使用换元积分法

= 25 s^{3} \cdot \int \frac{u sin ^{3} ( u )}{25 c ^{2} o ^{2}} d u

提出常数:

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

= 25 s^{3} \frac{1}{25 c ^{2} o ^{2}} \cdot \int u sin^{3} (u) d u

使用分布积分法

= 25 s^{3} \frac{1}{25 c ^{2} o ^{2}} (u (- cos (u) + \frac{1}{3} cos^{3} (u)) - \int - cos (u) + \frac{1}{3} cos^{3} (u) d u)

\int - cos (u) + \frac{1}{3} cos^{3} (u) d u = - \frac{2}{3} sin (u) - \frac{1}{9} sin^{3} (u)

= 25 s^{3} \frac{1}{25 c ^{2} o ^{2}} (u (- cos (u) + \frac{1}{3} cos^{3} (u)) - (- \frac{2}{3} sin (u) - \frac{1}{9} sin^{3} (u)))

u = 5 co x

代回

= 25 s^{3} \frac{1}{25 c ^{2} o ^{2}} (5 co x (- cos (5 co x) + \frac{1}{3} cos^{3} (5 co x)) - (- \frac{2}{3} sin (5 co x) - \frac{1}{9} sin^{3} (5 co x)))

化简

25 s^{3} \frac{1}{25 c ^{2} o ^{2}} (5 co x (- cos (5 co x) + \frac{1}{3} cos^{3} (5 co x)) - (- \frac{2}{3} sin (5 co x) - \frac{1}{9} sin^{3} (5 co x))) : s^{3} \frac{1}{c ^{2} o ^{2}} (- 5 co x cos (5 co x) + \frac{5}{3} co x cos^{3} (5 co x) + \frac{2}{3} sin (5 co x) + \frac{1}{9} sin^{3} (5 co x))

= s^{3} \frac{1}{c ^{2} o ^{2}} (- 5 co x cos (5 co x) + \frac{5}{3} co x cos^{3} (5 co x) + \frac{2}{3} sin (5 co x) + \frac{1}{9} sin^{3} (5 co x))

解答补常数

= s^{3} \frac{1}{c ^{2} o ^{2}} (- 5 co x cos (5 co x) + \frac{5}{3} co x cos^{3} (5 co x) + \frac{2}{3} sin (5 co x) + \frac{1}{9} sin^{3} (5 co x)) + C

作图

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