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(sqrt2-2sin(4x))^2

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

Limits of integration:

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The graph:

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Piecewise:

The solution

You have entered [src]
 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
  1. There are multiple ways to do this integral.

    Method #1

    1. Rewrite the integrand:

    2. Integrate term-by-term:

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

        1. Rewrite the integrand:

        2. Integrate term-by-term:

          1. The integral of a constant is the constant times the variable of integration:

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

            1. Let .

              Then let and substitute :

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

                1. The integral of cosine is sine:

                So, the result is:

              Now substitute back in:

            So, the result is:

          The result is:

        So, the result is:

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

        1. Let .

          Then let and substitute :

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

            1. The integral of sine is negative cosine:

            So, the result is:

          Now substitute back in:

        So, the result is:

      1. The integral of a constant is the constant times the variable of integration:

      The result is:

    Method #2

    1. Rewrite the integrand:

    2. Integrate term-by-term:

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

        1. Rewrite the integrand:

        2. Integrate term-by-term:

          1. The integral of a constant is the constant times the variable of integration:

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

            1. Let .

              Then let and substitute :

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

                1. The integral of cosine is sine:

                So, the result is:

              Now substitute back in:

            So, the result is:

          The result is:

        So, the result is:

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

        1. Let .

          Then let and substitute :

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

            1. The integral of sine is negative cosine:

            So, the result is:

          Now substitute back in:

        So, the result is:

      1. The integral of a constant is the constant times the variable of integration:

      The result is:

  2. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                              
 |                                                               
 |                     2                                         
 | /  ___             \                 sin(8*x)     ___         
 | \\/ 2  - 2*sin(4*x)/  dx = C + 4*x - -------- + \/ 2 *cos(4*x)
 |                                         4                     
/                                                                
$${{4\,x-{{\sin \left(8\,x\right)}\over{2}}}\over{2}}+\sqrt{2}\,\cos \left(4\,x\right)+2\,x$$
The graph
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  
$$- \frac{\sqrt{3}}{8} + \frac{3 \sqrt{2}}{2} + \frac{4 \pi}{3}$$
Numerical answer [src]
6.09360419739992
6.09360419739992
The graph
Integral of (sqrt2-2sin(4x))^2 dx

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