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Derivative of:
sin(2*x)^(2)
Identical expressions
sin(two *x)^(two)
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sinus of (two multiply by x) to the power of (two)
sin(2*x)(2)
sin2*x2
sin(2x)^(2)
sin(2x)(2)
sin2x2
sin2x^2
sin(2*x)^(2)dx
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Integral of sin(2*x)^(2) dx

The solution

You have entered [src]

 pi             
 --             
 2              
  /             
 |              
 |     2        
 |  sin (2*x) dx
 |              
/               
0

$$\int\limits_{0}^{\frac{\pi}{2}} \sin^{2}{\left(2 x \right)}\, dx$$

Integral(sin(2*x)^2, (x, 0, pi/2))

Detail solution

Rewrite the integrand:

$\sin^{2}{\left(2 x \right)} = \frac{1}{2} - \frac{\cos{\left(4 x \right)}}{2}$
Integrate term-by-term:
1. The integral of a constant is the constant times the variable of integration:
  
  $\int \frac{1}{2}\, dx = \frac{x}{2}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{\cos{\left(4 x \right)}}{2}\right)\, dx = - \frac{\int \cos{\left(4 x \right)}\, dx}{2}$
  1. Let $u = 4 x$ .
    
    Then let $du = 4 dx$ and substitute $\frac{du}{4}$ :
    
    $\int \frac{\cos{\left(u \right)}}{4}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{4}$
      1. The integral of cosine is sine:
        
        $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      So, the result is: $\frac{\sin{\left(u \right)}}{4}$
    Now substitute $u$ back in:
    
    $\frac{\sin{\left(4 x \right)}}{4}$
  So, the result is: $- \frac{\sin{\left(4 x \right)}}{8}$
The result is: $\frac{x}{2} - \frac{\sin{\left(4 x \right)}}{8}$
Add the constant of integration:

$\frac{x}{2} - \frac{\sin{\left(4 x \right)}}{8}+ \mathrm{constant}$

The answer is:

$\frac{x}{2} - \frac{\sin{\left(4 x \right)}}{8}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                               
 |                                
 |    2               x   sin(4*x)
 | sin (2*x) dx = C + - - --------
 |                    2      8    
/

$$\int \sin^{2}{\left(2 x \right)}\, dx = C + \frac{x}{2} - \frac{\sin{\left(4 x \right)}}{8}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

pi
--
4

$$\frac{\pi}{4}$$

pi
--
4

$$\frac{\pi}{4}$$

pi/4

Numerical answer [src]

0.785398163397448

0.785398163397448

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

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Integral of sin(2*x)^(2) dx

Limits of integration:

The graph:

Piecewise:

The solution