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

解答

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

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

求解步骤

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

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 = sin (x)

代回

= \frac{sin ^{5} ( x )}{5} - \frac{sin ^{7} ( x )}{7}

解答补常数

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

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