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受欢迎的微积分 >

integral of cos^5(t)

解答

\int cos^{5} (t) d t

解答

sin (t) - \frac{2 sin ^{3} ( t )}{3} + \frac{sin ^{5} ( t )}{5} + C

求解步骤

\int cos^{5} (t) d t

使用三角恒等式改写

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

使用换元积分法

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

乘开

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

= \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 = sin (t)

代回

= sin (t) - \frac{2 sin ^{3} ( t )}{3} + \frac{sin ^{5} ( t )}{5}

解答补常数

= sin (t) - \frac{2 sin ^{3} ( t )}{3} + \frac{sin ^{5} ( t )}{5} + C

作图

流行的例子

泰勒 cos(pi-6x)

t a y l or cos (π - 6 x)

derivative y=2sin(x)+ce^{-x}cos(x)

d er i v a t i v e y = 2 sin (x) + c e^{- x} cos (x)

18y^{''}+9y^'-2y=0

18 y^{^{''}} + 9 y^{^{'}} - 2 y = 0

y^{^{'}} = 3 x^{2} + 1

inverselaplace (5s-2)/(7s^2-s-11)

in v erse l a pl a ce \frac{5 s - 2}{7 s ^{2} - s - 11}