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sin(3x)^4

Integral of sin(3x)^4 dx

Limits of integration:

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The graph:

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Piecewise:

The solution

You have entered [src]
  1             
  /             
 |              
 |     4        
 |  sin (3*x) dx
 |              
/               
0               
$$\int\limits_{0}^{1} \sin^{4}{\left(3 x \right)}\, dx$$
Integral(sin(3*x)^4, (x, 0, 1))
Detail solution
  1. Rewrite the integrand:

  2. 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 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:

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

          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 cosine is sine:

            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 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:

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

          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 cosine is sine:

            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:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                             
 |                                              
 |    4               sin(6*x)   sin(12*x)   3*x
 | sin (3*x) dx = C - -------- + --------- + ---
 |                       12          96       8 
/                                               
$${{{{{{\sin \left(12\,x\right)}\over{2}}+6\,x}\over{8}}-{{\sin \left(6\,x\right)}\over{2}}+{{3\,x}\over{2}}}\over{6}}$$
The graph
The answer [src]
                       3          
3   cos(3)*sin(3)   sin (3)*cos(3)
- - ------------- - --------------
8         8               12      
$${{\sin 12-8\,\sin 6+36}\over{96}}$$
=
=
                       3          
3   cos(3)*sin(3)   sin (3)*cos(3)
- - ------------- - --------------
8         8               12      
$$- \frac{\sin^{3}{\left(3 \right)} \cos{\left(3 \right)}}{12} - \frac{\sin{\left(3 \right)} \cos{\left(3 \right)}}{8} + \frac{3}{8}$$
Numerical answer [src]
0.392695323620739
0.392695323620739
The graph
Integral of sin(3x)^4 dx

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