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受欢迎的微积分 >

integral of cos^7(y)

解答

\int cos^{7} (y) d y

解答

sin (y) - sin^{3} (y) + \frac{3 sin ^{5} ( y )}{5} - \frac{sin ^{7} ( y )}{7} + C

求解步骤

\int cos^{7} (y) d y

使用三角恒等式改写

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

使用换元积分法

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

乘开

(1 - u^{2})^{3} : 1 - 3 u^{2} + 3 u^{4} - u^{6}

= \int 1 - 3 u^{2} + 3 u^{4} - u^{6} d u

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 1 d u - \int 3 u^{2} d u + \int 3 u^{4} d u - \int u^{6} d u

\int 1 d u = u

\int 3 u^{2} d u = u^{3}

\int 3 u^{4} d u = \frac{3 u ^{5}}{5}

\int u^{6} d u = \frac{u ^{7}}{7}

= u - u^{3} + \frac{3 u ^{5}}{5} - \frac{u ^{7}}{7}

u = sin (y)

代回

= sin (y) - sin^{3} (y) + \frac{3 sin ^{5} ( y )}{5} - \frac{sin ^{7} ( y )}{7}

解答补常数

= sin (y) - sin^{3} (y) + \frac{3 sin ^{5} ( y )}{5} - \frac{sin ^{7} ( y )}{7} + C

作图

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(dx)/(dt)-0.06x=200

\frac{d x}{d t} - 0.06 x = 200

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