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受欢迎的微积分 >

integral of (2+cos(2x))^2

解答

\int (2 + cos (2 x))^{2} d x

解答

\frac{1}{8} (36 x + 16 sin (2 x) + sin (4 x)) + C

求解步骤

\int (2 + cos (2 x))^{2} d x

使用换元积分法

= \int (2 + cos (u))^{2} \frac{1}{2} d u

提出常数:

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

= \frac{1}{2} \cdot \int (2 + cos (u))^{2} d u

乘开

(2 + cos (u))^{2} : 4 + 4 cos (u) + cos^{2} (u)

= \frac{1}{2} \cdot \int 4 + 4 cos (u) + cos^{2} (u) 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

= \frac{1}{2} (\int 4 d u + \int 4 cos (u) d u + \int cos^{2} (u) d u)

\int 4 d u = 4 u

\int 4 cos (u) d u = 4 sin (u)

\int cos^{2} (u) d u = \frac{1}{2} (u + \frac{1}{2} sin (2 u))

= \frac{1}{2} (4 u + 4 sin (u) + \frac{1}{2} (u + \frac{1}{2} sin (2 u)))

u = 2 x

代回

= \frac{1}{2} (4 \cdot 2 x + 4 sin (2 x) + \frac{1}{2} (2 x + \frac{1}{2} sin (2 \cdot 2 x)))

化简

\frac{1}{2} (4 \cdot 2 x + 4 sin (2 x) + \frac{1}{2} (2 x + \frac{1}{2} sin (2 \cdot 2 x))) : \frac{1}{8} (36 x + 16 sin (2 x) + sin (4 x))

= \frac{1}{8} (36 x + 16 sin (2 x) + sin (4 x))

解答补常数

= \frac{1}{8} (36 x + 16 sin (2 x) + sin (4 x)) + C

作图

流行的例子

integral from-1 to 2 of (2-x^2)-(-x)

\int_{- 1}^{2} (2 - x^{2}) - (- x) d x

(\partial)/(\partial y)(-0.27y^2+3.78y)

\frac{\partial}{\partial y} (- 0.27 y^{2} + 3.78 y)

derivative y=(f(x))/(x^4)

d er i v a t i v e y = \frac{f ( x )}{x ^{4}}

derivative of e^{x^2}-cos(x)

\frac{d}{d x} (e^{x^{2}} - cos (x))

integral of 4/(x^5)

\int \frac{4}{x ^{5}} d x