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

解答

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

- cos (x) + \frac{2 cos ^{3} ( x )}{3} - \frac{cos ^{5} ( x )}{5} + C

求解步骤

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

化简

sin^{4} (x) sin (x) : sin^{5} (x)

= \int sin^{5} (x) d x

使用三角恒等式改写

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

使用换元积分法

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

\int 1 d u = u

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

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

= - u + \frac{2 u ^{3}}{3} - \frac{u ^{5}}{5}

u = cos (x)

代回

= - cos (x) + \frac{2 cos ^{3} ( x )}{3} - \frac{cos ^{5} ( x )}{5}

解答补常数

= - cos (x) + \frac{2 cos ^{3} ( x )}{3} - \frac{cos ^{5} ( x )}{5} + C

\int 2 + 4 - x d x

\int 5 sinh (ln (2) - \frac{x}{3}) d x

\int \frac{sin ( 2 x )}{e ^{- 4 x}} d x

\int (5 x + 7)^{5} d x

\int lo g_{e} (3 x) d x