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sin2x*sinx*cos3x

Integral of sin2x*sinx*cos3x dx

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The graph:

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Piecewise:

The solution

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 |  sin(2*x)*sin(x)*cos(3*x) dx
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$$\int\limits_{0}^{1} \sin{\left(x \right)} \sin{\left(2 x \right)} \cos{\left(3 x \right)}\, dx$$
Integral(sin(2*x)*sin(x)*cos(3*x), (x, 0, 1))
Detail solution
  1. There are multiple ways to do this integral.

    Method #1

    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. Rewrite the integrand:

          2. Let .

            Then let and substitute :

            1. 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. Let .

                  Then let and substitute :

                  1. 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. The integral of is when :

                      So, the result is:

                    The result is:

                  Now substitute back in:

                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 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. The integral of cosine is sine:

                So, the result is:

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

              The result is:

            Now substitute back in:

          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. Let .

            Then let and substitute :

            1. 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. 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:

              The result is:

            Now substitute back in:

          So, the result is:

        The result is:

      So, 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. Let .

          Then let and substitute :

          1. 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. Let .

                Then let and substitute :

                1. 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. The integral of is when :

                    So, the result is:

                  The result is:

                Now substitute back in:

              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 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. The integral of cosine is sine:

              So, the result is:

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

            The result is:

          Now substitute back in:

        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. Let .

          Then let and substitute :

          1. 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. 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:

            The result is:

          Now substitute back in:

        So, the result is:

      The result is:

  2. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                         3                
 |                                   x   sin (2*x)   sin(4*x)
 | sin(2*x)*sin(x)*cos(3*x) dx = C - - + --------- + --------
 |                                   4       6          16   
/                                                            
$$\int \sin{\left(x \right)} \sin{\left(2 x \right)} \cos{\left(3 x \right)}\, dx = C - \frac{x}{4} + \frac{\sin^{3}{\left(2 x \right)}}{6} + \frac{\sin{\left(4 x \right)}}{16}$$
The graph
The answer [src]
  cos(1)*cos(2)*cos(3)   cos(1)*sin(2)*sin(3)   cos(2)*sin(1)*sin(3)   cos(1)*cos(3)*sin(2)   cos(3)*sin(1)*sin(2)   cos(1)*cos(2)*sin(3)   5*sin(1)*sin(2)*sin(3)
- -------------------- - -------------------- - -------------------- - -------------------- + -------------------- + -------------------- + ----------------------
           4                      4                      4                      8                      4                      6                       24          
$$\frac{\sin{\left(1 \right)} \sin{\left(2 \right)} \cos{\left(3 \right)}}{4} - \frac{\cos{\left(1 \right)} \cos{\left(2 \right)} \cos{\left(3 \right)}}{4} - \frac{\sin{\left(2 \right)} \sin{\left(3 \right)} \cos{\left(1 \right)}}{4} + \frac{\sin{\left(3 \right)} \cos{\left(1 \right)} \cos{\left(2 \right)}}{6} - \frac{\sin{\left(1 \right)} \sin{\left(3 \right)} \cos{\left(2 \right)}}{4} + \frac{5 \sin{\left(1 \right)} \sin{\left(2 \right)} \sin{\left(3 \right)}}{24} - \frac{\sin{\left(2 \right)} \cos{\left(1 \right)} \cos{\left(3 \right)}}{8}$$
=
=
  cos(1)*cos(2)*cos(3)   cos(1)*sin(2)*sin(3)   cos(2)*sin(1)*sin(3)   cos(1)*cos(3)*sin(2)   cos(3)*sin(1)*sin(2)   cos(1)*cos(2)*sin(3)   5*sin(1)*sin(2)*sin(3)
- -------------------- - -------------------- - -------------------- - -------------------- + -------------------- + -------------------- + ----------------------
           4                      4                      4                      8                      4                      6                       24          
$$\frac{\sin{\left(1 \right)} \sin{\left(2 \right)} \cos{\left(3 \right)}}{4} - \frac{\cos{\left(1 \right)} \cos{\left(2 \right)} \cos{\left(3 \right)}}{4} - \frac{\sin{\left(2 \right)} \sin{\left(3 \right)} \cos{\left(1 \right)}}{4} + \frac{\sin{\left(3 \right)} \cos{\left(1 \right)} \cos{\left(2 \right)}}{6} - \frac{\sin{\left(1 \right)} \sin{\left(3 \right)} \cos{\left(2 \right)}}{4} + \frac{5 \sin{\left(1 \right)} \sin{\left(2 \right)} \sin{\left(3 \right)}}{24} - \frac{\sin{\left(2 \right)} \cos{\left(1 \right)} \cos{\left(3 \right)}}{8}$$
Numerical answer [src]
-0.17199566517858
-0.17199566517858
The graph
Integral of sin2x*sinx*cos3x dx

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