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受欢迎的微积分 >

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

解答

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

解答

\frac{1}{2} (- cot (2 x) - csc (2 x)) + C

求解步骤

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

使用换元积分法

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

乘开

\frac{1 + cos ( u )}{sin ^{2} ( u )} : \frac{1}{sin ^{2} ( u )} + \frac{cos ( u )}{sin ^{2} ( u )}

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

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

\int \frac{cos ( u )}{sin ^{2} ( u )} d u = - csc (u)

= \frac{1}{2} (- cot (u) - csc (u))

u = 2 x

代回

= \frac{1}{2} (- cot (2 x) - csc (2 x))

解答补常数

= \frac{1}{2} (- cot (2 x) - csc (2 x)) + C

作图

流行的例子

(\partial)/(\partial y)(x+3y)

\frac{\partial}{\partial y} (x + 3 y)

斜率 (-1.3)(3.5)

s l o p e (- 1.3) (3.5)

derivative g(x)=ln^3(x)

d er i v a t i v e g (x) = ln^{3} (x)

tangent y=(5x)/(2^x),\at x=2

t an g e n t y = \frac{5 x}{2 ^{x}}, at x = 2

(\partial)/(\partial y)(ln(x^2+3y))

\frac{\partial}{\partial y} (ln (x^{2} + 3 y))