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受欢迎的微积分 >

integral of sin^{18}(x)cos^3(x)

解答

\int sin^{18} (x) cos^{3} (x) d x

解答

\frac{sin ^{19} ( x )}{19} - \frac{sin ^{21} ( x )}{21} + C

求解步骤

\int sin^{18} (x) cos^{3} (x) d x

使用三角恒等式改写

= \int (1 - sin^{2} (x)) cos (x) sin^{18} (x) d x

使用换元积分法

= \int u^{18} (1 - u^{2}) d u

乘开

u^{18} (1 - u^{2}) : u^{18} - u^{20}

= \int u^{18} - u^{20} 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 u^{18} d u - \int u^{20} d u

\int u^{18} d u = \frac{u ^{19}}{19}

\int u^{20} d u = \frac{u ^{21}}{21}

= \frac{u ^{19}}{19} - \frac{u ^{21}}{21}

u = sin (x)

代回

= \frac{sin ^{19} ( x )}{19} - \frac{sin ^{21} ( x )}{21}

解答补常数

= \frac{sin ^{19} ( x )}{19} - \frac{sin ^{21} ( x )}{21} + C

作图

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