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

解答

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

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

求解步骤

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

使用三角恒等式改写

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

代回

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

解答补常数

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

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