Integral of ycos2xy dx

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  2                
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 |  y*cos(2*x)*y dx
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1

$$\int\limits_{1}^{2} y y \cos{\left(2 x \right)}\, dx$$

Integral((y*cos(2*x))*y, (x, 1, 2))

Detail solution

The integral of a constant times a function is the constant times the integral of the function:

$\int y y \cos{\left(2 x \right)}\, dx = y \int y \cos{\left(2 x \right)}\, dx$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int y \cos{\left(2 x \right)}\, dx = y \int \cos{\left(2 x \right)}\, dx$
  1. Let $u = 2 x$ .
    
    Then let $du = 2 dx$ 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:
    
    $\frac{\sin{\left(2 x \right)}}{2}$
  So, the result is: $\frac{y \sin{\left(2 x \right)}}{2}$
So, the result is: $\frac{y^{2} \sin{\left(2 x \right)}}{2}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                       2         
 |                       y *sin(2*x)
 | y*cos(2*x)*y dx = C + -----------
 |                            2     
/

$$\int y y \cos{\left(2 x \right)}\, dx = C + \frac{y^{2} \sin{\left(2 x \right)}}{2}$$

Simplify

The answer [src]

 2           2       
y *sin(4)   y *sin(2)
--------- - ---------
    2           2

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

 2           2       
y *sin(4)   y *sin(2)
--------- - ---------
    2           2

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

y^2*sin(4)/2 - y^2*sin(2)/2

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