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

Limits of integration:

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The graph:

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Piecewise:

The solution

You have entered [src]
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 |  cos(x)*sin(3*x) dx
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$$\int\limits_{0}^{0} \sin{\left(3 x \right)} \cos{\left(x \right)}\, dx$$
Integral(cos(x)*sin(3*x), (x, 0, 0))
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. There are multiple ways to do this integral.

        Method #1

        1. Let .

          Then let and substitute :

          1. The integral of is when :

          Now substitute back in:

        Method #2

        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. The integral of is when :

              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. 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 is when :

          So, the result is:

        Now substitute back in:

      So, the result is:

    The result is:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                        2   
 |                             4      3*cos (x)
 | cos(x)*sin(3*x) dx = C - sin (x) - ---------
 |                                        2    
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$$\int \sin{\left(3 x \right)} \cos{\left(x \right)}\, dx = C - \sin^{4}{\left(x \right)} - \frac{3 \cos^{2}{\left(x \right)}}{2}$$
The graph
The answer [src]
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Numerical answer [src]
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    Use the examples entering the upper and lower limits of integration.