积分 1-cos(2y) dx

解答

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 cos(y)                 
    /                   
   |                    
   |   (1 - cos(2*y)) dy
   |                    
  /                     
  0

$$\int\limits_{0}^{\cos{\left(y \right)}} \left(1 - \cos{\left(2 y \right)}\right)\, dy$$

Integral(1 - cos(2*y), (y, 0, cos(y)))

Detail solution

Integrate term-by-term:
1. The integral of a constant is the constant times the variable of integration:
  
  $\int 1\, dy = y$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \cos{\left(2 y \right)}\right)\, dy = - \int \cos{\left(2 y \right)}\, dy$
  1. Let $u = 2 y$ .
    
    Then let $du = 2 dy$ and substitute $\frac{du}{2}$ :
    
    $\int \frac{\cos{\left(u \right)}}{2}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{2}$
      1. The integral of cosine is sine:
        
        $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      So, the result is: $\frac{\sin{\left(u \right)}}{2}$
    Now substitute $u$ back in:
    
    $\frac{\sin{\left(2 y \right)}}{2}$
  So, the result is: $- \frac{\sin{\left(2 y \right)}}{2}$
The result is: $y - \frac{\sin{\left(2 y \right)}}{2}$
Add the constant of integration:

$y - \frac{\sin{\left(2 y \right)}}{2}+ \mathrm{constant}$

The answer is:

$y - \frac{\sin{\left(2 y \right)}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                    
 |                             sin(2*y)
 | (1 - cos(2*y)) dy = C + y - --------
 |                                2    
/

$$\int \left(1 - \cos{\left(2 y \right)}\right)\, dy = C + y - \frac{\sin{\left(2 y \right)}}{2}$$

Simplify

The answer [src]

  sin(2*cos(y))         
- ------------- + cos(y)
        2

$$- \frac{\sin{\left(2 \cos{\left(y \right)} \right)}}{2} + \cos{\left(y \right)}$$

  sin(2*cos(y))         
- ------------- + cos(y)
        2

$$- \frac{\sin{\left(2 \cos{\left(y \right)} \right)}}{2} + \cos{\left(y \right)}$$

-sin(2*cos(y))/2 + cos(y)

这些示例也可用于输入定积分的上下限.

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积分 1-cos(2y) dx

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