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受欢迎的微积分 >

integral of e^{-cos(x)}sin^3(x)

解答

\int e^{- c o s (x)} sin^{3} (x) d x

解答

- (- 2 (- e^{- c o s (x)} - e^{- c o s (x)} cos (x)) - e^{- c o s (x)} sin^{2} (x)) + C

求解步骤

\int e^{- c o s (x)} sin^{3} (x) d x

使用换元积分法

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

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

使用分布积分法

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

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

= - (- e^{- u} (1 - u^{2}) - 2 (- e^{- u} u - e^{- u}))

u = cos (x)

代回

= - (- e^{- c o s (x)} (1 - cos^{2} (x)) - 2 (- e^{- c o s (x)} cos (x) - e^{- c o s (x)}))

使用毕达哥拉斯恒等式:

1 = cos^{2} (x) + sin^{2} (x)

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

= - (- 2 (- e^{- c o s (x)} - e^{- c o s (x)} cos (x)) - e^{- c o s (x)} sin^{2} (x))

解答补常数

= - (- 2 (- e^{- c o s (x)} - e^{- c o s (x)} cos (x)) - e^{- c o s (x)} sin^{2} (x)) + C

作图

流行的例子

limit as x approaches 0 of cot(x)

x \to 0 lim (cot (x))

\frac{d}{d y} (2 y - 1)

y^{''}-2y^'+1y=0,y(0)=-1,y^'(0)=2

y^{^{''}} - 2 y^{^{'}} + 1 y = 0, y (0) = - 1, y^{^{'}} (0) = 2

integral of 1/(cos^2(x))

\int \frac{1}{cos ^{2} ( x )} d x

derivative of ln(x+x)

\frac{d}{d x} (ln (x) + x)