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受欢迎的微积分 >

integral of sin^7(xco)s^3x

解答

\int sin^{7} (x co) s^{3} x d x

解答

\frac{s ^{3}}{3675 c ^{2} o ^{2}} (- 3675 co x cos (co x) + 3675 co x cos^{3} (co x) - 2205 co x cos^{5} (co x) + 525 co x cos^{7} (co x) + 280 sin^{3} (co x) + 126 sin^{5} (co x) + 75 sin^{7} (co x) + 1680 sin (co x)) + C

求解步骤

\int sin^{7} (x co) s^{3} x d x

提出常数:

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

= s^{3} \cdot \int sin^{7} (co x) x d x

使用换元积分法

= s^{3} \cdot \int \frac{u sin ^{7} ( u )}{c ^{2} o ^{2}} d u

提出常数:

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

= s^{3} \frac{1}{c ^{2} o ^{2}} \cdot \int u sin^{7} (u) d u

使用分布积分法

= s^{3} \frac{1}{c ^{2} o ^{2}} (u (- cos (u) + cos^{3} (u) - \frac{3}{5} cos^{5} (u) + \frac{1}{7} cos^{7} (u)) - \int - cos (u) + cos^{3} (u) - \frac{3}{5} cos^{5} (u) + \frac{1}{7} cos^{7} (u) d u)

\int - cos (u) + cos^{3} (u) - \frac{3}{5} cos^{5} (u) + \frac{1}{7} cos^{7} (u) d u = - \frac{16}{35} sin (u) + \frac{9}{35} sin^{3} (u) - \frac{3}{25} sin^{5} (u) - \frac{1}{3} sin^{3} (u) + \frac{3}{35} sin^{5} (u) - \frac{1}{49} sin^{7} (u)

= s^{3} \frac{1}{c ^{2} o ^{2}} (u (- cos (u) + cos^{3} (u) - \frac{3}{5} cos^{5} (u) + \frac{1}{7} cos^{7} (u)) - (- \frac{16}{35} sin (u) + \frac{9}{35} sin^{3} (u) - \frac{3}{25} sin^{5} (u) - \frac{1}{3} sin^{3} (u) + \frac{3}{35} sin^{5} (u) - \frac{1}{49} sin^{7} (u)))

u = co x

代回

= s^{3} \frac{1}{c ^{2} o ^{2}} (co x (- cos (co x) + cos^{3} (co x) - \frac{3}{5} cos^{5} (co x) + \frac{1}{7} cos^{7} (co x)) - (- \frac{16}{35} sin (co x) + \frac{9}{35} sin^{3} (co x) - \frac{3}{25} sin^{5} (co x) - \frac{1}{3} sin^{3} (co x) + \frac{3}{35} sin^{5} (co x) - \frac{1}{49} sin^{7} (co x)))

化简

s^{3} \frac{1}{c ^{2} o ^{2}} (co x (- cos (co x) + cos^{3} (co x) - \frac{3}{5} cos^{5} (co x) + \frac{1}{7} cos^{7} (co x)) - (- \frac{16}{35} sin (co x) + \frac{9}{35} sin^{3} (co x) - \frac{3}{25} sin^{5} (co x) - \frac{1}{3} sin^{3} (co x) + \frac{3}{35} sin^{5} (co x) - \frac{1}{49} sin^{7} (co x))) : \frac{s ^{3}}{3675 c ^{2} o ^{2}} (- 3675 co x cos (co x) + 3675 co x cos^{3} (co x) - 2205 co x cos^{5} (co x) + 525 co x cos^{7} (co x) + 280 sin^{3} (co x) + 126 sin^{5} (co x) + 75 sin^{7} (co x) + 1680 sin (co x))

= \frac{s ^{3}}{3675 c ^{2} o ^{2}} (- 3675 co x cos (co x) + 3675 co x cos^{3} (co x) - 2205 co x cos^{5} (co x) + 525 co x cos^{7} (co x) + 280 sin^{3} (co x) + 126 sin^{5} (co x) + 75 sin^{7} (co x) + 1680 sin (co x))

解答补常数

= \frac{s ^{3}}{3675 c ^{2} o ^{2}} (- 3675 co x cos (co x) + 3675 co x cos^{3} (co x) - 2205 co x cos^{5} (co x) + 525 co x cos^{7} (co x) + 280 sin^{3} (co x) + 126 sin^{5} (co x) + 75 sin^{7} (co x) + 1680 sin (co x)) + C

作图

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