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受欢迎的微积分 >

integral of 7sin^7(x)

解答

\int 7 sin^{7} (x) d x

解答

- 7 cos (x) + 7 cos^{3} (x) - \frac{21}{5} cos^{5} (x) + cos^{7} (x) + C

求解步骤

\int 7 sin^{7} (x) d x

提出常数:

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

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

使用三角恒等式改写

= 7 \cdot \int (1 - cos^{2} (x))^{3} sin (x) d x

使用换元积分法

= 7 \cdot \int - (1 - u^{2})^{3} d u

乘开

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

= 7 \cdot \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

= 7 (- \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}

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

u = cos (x)

代回

= 7 (- cos (x) + cos^{3} (x) - \frac{3 cos ^{5} ( x )}{5} + \frac{cos ^{7} ( x )}{7})

化简

7 (- cos (x) + cos^{3} (x) - \frac{3 cos ^{5} ( x )}{5} + \frac{cos ^{7} ( x )}{7}) : - 7 cos (x) + 7 cos^{3} (x) - \frac{21}{5} cos^{5} (x) + cos^{7} (x)

= - 7 cos (x) + 7 cos^{3} (x) - \frac{21}{5} cos^{5} (x) + cos^{7} (x)

解答补常数

= - 7 cos (x) + 7 cos^{3} (x) - \frac{21}{5} cos^{5} (x) + cos^{7} (x) + C

作图

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