integral of sin^3(r)cos^4(r)

解答

\int sin^{3} (r) cos^{4} (r) d r

- \frac{cos ^{5} ( r )}{5} + \frac{cos ^{7} ( r )}{7} + C

求解步骤

\int sin^{3} (r) cos^{4} (r) d r

使用三角恒等式改写

= \int (1 - cos^{2} (r)) sin (r) cos^{4} (r) d r

使用换元积分法

= \int - u^{4} (1 - u^{2}) d u

乘开

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

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

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

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

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

u = cos (r)

代回

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

解答补常数

= - \frac{cos ^{5} ( r )}{5} + \frac{cos ^{7} ( r )}{7} + C

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