Integral of sin^2y dy

The solution

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  1           
  /           
 |            
 |     2      
 |  sin (y) dy
 |            
/             
0

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

Integral(sin(y)^2, (y, 0, 1))

Detail solution

Rewrite the integrand:

$\sin^{2}{\left(y \right)} = \frac{1}{2} - \frac{\cos{\left(2 y \right)}}{2}$
Integrate term-by-term:
1. The integral of a constant is the constant times the variable of integration:
  
  $\int \frac{1}{2}\, dy = \frac{y}{2}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{\cos{\left(2 y \right)}}{2}\right)\, dy = - \frac{\int \cos{\left(2 y \right)}\, dy}{2}$
  1. Let $u = 2 y$ .
    
    Then let $du = 2 dy$ and substitute $\frac{du}{2}$ :
    
    $\int \frac{\cos{\left(u \right)}}{4}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\cos{\left(u \right)}}{2}\, du = \frac{\int \cos{\left(u \right)}\, du}{2}$
      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: $- \frac{\sin{\left(2 y \right)}}{4}$
The result is: $\frac{y}{2} - \frac{\sin{\left(2 y \right)}}{4}$
Add the constant of integration:

$\frac{y}{2} - \frac{\sin{\left(2 y \right)}}{4}+ \mathrm{constant}$

The answer is:

$\frac{y}{2} - \frac{\sin{\left(2 y \right)}}{4}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                             
 |                              
 |    2             y   sin(2*y)
 | sin (y) dy = C + - - --------
 |                  2      4    
/

$${{y-{{\sin \left(2\,y\right)}\over{2}}}\over{2}}$$

Simplify

The answer [src]

1   cos(1)*sin(1)
- - -------------
2         2

$$-{{\sin 2-2}\over{4}}$$

1   cos(1)*sin(1)
- - -------------
2         2

$$- \frac{\sin{\left(1 \right)} \cos{\left(1 \right)}}{2} + \frac{1}{2}$$

Упростить

Numerical answer [src]

0.27267564329358

0.27267564329358

Use the examples entering the upper and lower limits of integration.

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Integral of sin^2y dy

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The solution