integral of sin^2(x)sin(x)

解答

\int sin^{2} (x) sin (x) d x

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

求解步骤

\int sin^{2} (x) sin (x) d x

使用三角恒等式改写

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

使用换元积分法

= \int - 1 + u^{2} 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 1 d u + \int u^{2} d u

\int 1 d u = u

\int u^{2} d u = \frac{u ^{3}}{3}

= - u + \frac{u ^{3}}{3}

u = cos (x)

代回

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

解答补常数

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