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sin(3*x)*sin(x)

Integral of sin(3*x)*sin(x) dx

Limits of integration:

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The graph:

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Piecewise:

The solution

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 |  sin(3*x)*sin(x) dx
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$$\int\limits_{0}^{1} \sin{\left(x \right)} \sin{\left(3 x \right)}\, dx$$
Integral(sin(3*x)*sin(x), (x, 0, 1))
Detail solution
  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. 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:

      So, the result is:

    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:

    The result is:

  3. Now simplify:

  4. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                            
 |                          sin(4*x)   sin(2*x)
 | sin(3*x)*sin(x) dx = C - -------- + --------
 |                             8          4    
/                                              
$$\int \sin{\left(x \right)} \sin{\left(3 x \right)}\, dx = C + \frac{\sin{\left(2 x \right)}}{4} - \frac{\sin{\left(4 x \right)}}{8}$$
The graph
The answer [src]
  3*cos(3)*sin(1)   cos(1)*sin(3)
- --------------- + -------------
         8                8      
$$\frac{\sin{\left(3 \right)} \cos{\left(1 \right)}}{8} - \frac{3 \sin{\left(1 \right)} \cos{\left(3 \right)}}{8}$$
=
=
  3*cos(3)*sin(1)   cos(1)*sin(3)
- --------------- + -------------
         8                8      
$$\frac{\sin{\left(3 \right)} \cos{\left(1 \right)}}{8} - \frac{3 \sin{\left(1 \right)} \cos{\left(3 \right)}}{8}$$
-3*cos(3)*sin(1)/8 + cos(1)*sin(3)/8
Numerical answer [src]
0.321924668619911
0.321924668619911
The graph
Integral of sin(3*x)*sin(x) dx

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