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Identical expressions
(sqrt two -2sin(4x))^2
( square root of 2 minus 2 sinus of (4x)) squared
( square root of two minus 2 sinus of (4x)) squared
(√2-2sin(4x))^2
(sqrt2-2sin(4x))2
sqrt2-2sin4x2
(sqrt2-2sin(4x))²
(sqrt2-2sin(4x)) to the power of 2
sqrt2-2sin4x^2
(sqrt2-2sin(4x))^2dx
Similar expressions
(sqrt2+2sin(4x))^2

Solving integrals/ (sqrt2-2sin(4x))^2

Integral of (sqrt2-2sin(4x))^2 dx

The solution

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 pi                         
 --                         
 2                          
  /                         
 |                          
 |                      2   
 |  /  ___             \    
 |  \\/ 2  - 2*sin(4*x)/  dx
 |                          
/                           
pi                          
--                          
6

$$\int\limits_{\frac{\pi}{6}}^{\frac{\pi}{2}} \left(- 2 \sin{\left(4 x \right)} + \sqrt{2}\right)^{2}\, dx$$

Integral((sqrt(2) - 2*sin(4*x))^2, (x, pi/6, pi/2))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\left(- 2 \sin{\left(4 x \right)} + \sqrt{2}\right)^{2} = 4 \sin^{2}{\left(4 x \right)} - 4 \sqrt{2} \sin{\left(4 x \right)} + 2$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 4 \sin^{2}{\left(4 x \right)}\, dx = 4 \int \sin^{2}{\left(4 x \right)}\, dx$
    1. Rewrite the integrand:
      
      $\sin^{2}{\left(4 x \right)} = \frac{1}{2} - \frac{\cos{\left(8 x \right)}}{2}$
    2. Integrate term-by-term:
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int \frac{1}{2}\, dx = \frac{x}{2}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{\cos{\left(8 x \right)}}{2}\right)\, dx = - \frac{\int \cos{\left(8 x \right)}\, dx}{2}$
      
      Let $u = 8 x$ .
      
      Then let $du = 8 dx$ and substitute $\frac{du}{8}$ :
      
      $\int \frac{\cos{\left(u \right)}}{64}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\cos{\left(u \right)}}{8}\, du = \frac{\int \cos{\left(u \right)}\, du}{8}$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      So, the result is: $\frac{\sin{\left(u \right)}}{8}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin{\left(8 x \right)}}{8}$
      
      So, the result is: $- \frac{\sin{\left(8 x \right)}}{16}$
      
      The result is: $\frac{x}{2} - \frac{\sin{\left(8 x \right)}}{16}$
    So, the result is: $2 x - \frac{\sin{\left(8 x \right)}}{4}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 4 \sqrt{2} \sin{\left(4 x \right)}\right)\, dx = - 4 \sqrt{2} \int \sin{\left(4 x \right)}\, dx$
    1. Let $u = 4 x$ .
      
      Then let $du = 4 dx$ and substitute $\frac{du}{4}$ :
      
      $\int \frac{\sin{\left(u \right)}}{16}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\sin{\left(u \right)}}{4}\, du = \frac{\int \sin{\left(u \right)}\, du}{4}$
      
      The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
      
      So, the result is: $- \frac{\cos{\left(u \right)}}{4}$
      
      Now substitute $u$ back in:
      
      $- \frac{\cos{\left(4 x \right)}}{4}$
    So, the result is: $\sqrt{2} \cos{\left(4 x \right)}$
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int 2\, dx = 2 x$
  The result is: $4 x - \frac{\sin{\left(8 x \right)}}{4} + \sqrt{2} \cos{\left(4 x \right)}$
Method #2
1. Rewrite the integrand:
  
  $\left(- 2 \sin{\left(4 x \right)} + \sqrt{2}\right)^{2} = 4 \sin^{2}{\left(4 x \right)} - 4 \sqrt{2} \sin{\left(4 x \right)} + 2$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 4 \sin^{2}{\left(4 x \right)}\, dx = 4 \int \sin^{2}{\left(4 x \right)}\, dx$
    1. Rewrite the integrand:
      
      $\sin^{2}{\left(4 x \right)} = \frac{1}{2} - \frac{\cos{\left(8 x \right)}}{2}$
    2. Integrate term-by-term:
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int \frac{1}{2}\, dx = \frac{x}{2}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{\cos{\left(8 x \right)}}{2}\right)\, dx = - \frac{\int \cos{\left(8 x \right)}\, dx}{2}$
      
      Let $u = 8 x$ .
      
      Then let $du = 8 dx$ and substitute $\frac{du}{8}$ :
      
      $\int \frac{\cos{\left(u \right)}}{64}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\cos{\left(u \right)}}{8}\, du = \frac{\int \cos{\left(u \right)}\, du}{8}$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      So, the result is: $\frac{\sin{\left(u \right)}}{8}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin{\left(8 x \right)}}{8}$
      
      So, the result is: $- \frac{\sin{\left(8 x \right)}}{16}$
      
      The result is: $\frac{x}{2} - \frac{\sin{\left(8 x \right)}}{16}$
    So, the result is: $2 x - \frac{\sin{\left(8 x \right)}}{4}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 4 \sqrt{2} \sin{\left(4 x \right)}\right)\, dx = - 4 \sqrt{2} \int \sin{\left(4 x \right)}\, dx$
    1. Let $u = 4 x$ .
      
      Then let $du = 4 dx$ and substitute $\frac{du}{4}$ :
      
      $\int \frac{\sin{\left(u \right)}}{16}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\sin{\left(u \right)}}{4}\, du = \frac{\int \sin{\left(u \right)}\, du}{4}$
      
      The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
      
      So, the result is: $- \frac{\cos{\left(u \right)}}{4}$
      
      Now substitute $u$ back in:
      
      $- \frac{\cos{\left(4 x \right)}}{4}$
    So, the result is: $\sqrt{2} \cos{\left(4 x \right)}$
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int 2\, dx = 2 x$
  The result is: $4 x - \frac{\sin{\left(8 x \right)}}{4} + \sqrt{2} \cos{\left(4 x \right)}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

$${{4\,x-{{\sin \left(8\,x\right)}\over{2}}}\over{2}}+\sqrt{2}\,\cos \left(4\,x\right)+2\,x$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

    ___       ___       
  \/ 3    3*\/ 2    4*pi
- ----- + ------- + ----
    8        2       3

$${{3\,\sin \left({{4\,\pi}\over{3}}\right)-8\,\pi-3\,2^{{{5}\over{2 }}}\,\cos \left({{2\,\pi}\over{3}}\right)}\over{12}}-{{\sin \left(4 \,\pi\right)-2^{{{5}\over{2}}}\,\cos \left(2\,\pi\right)-8\,\pi }\over{4}}$$

    ___       ___       
  \/ 3    3*\/ 2    4*pi
- ----- + ------- + ----
    8        2       3

$$- \frac{\sqrt{3}}{8} + \frac{3 \sqrt{2}}{2} + \frac{4 \pi}{3}$$

Упростить

Numerical answer [src]

6.09360419739992

6.09360419739992

The graph

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

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Identical expressions

Similar expressions

Integral of (sqrt2-2sin(4x))^2 dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2