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sin(5x)*cos(4x)

Integral of sin(5x)*cos(4x) dx

Limits of integration:

from to
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The graph:

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Piecewise:

The solution

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  1                     
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 |  sin(5*x)*cos(4*x) dx
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$$\int\limits_{0}^{1} \sin{\left(5 x \right)} \cos{\left(4 x \right)}\, dx$$
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. 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 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 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:

        Method #2

        1. Rewrite the integrand:

        2. Integrate term-by-term:

          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:

          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:

          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:

          The result is:

        Method #3

        1. Rewrite the integrand:

        2. Integrate term-by-term:

          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:

          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:

          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:

          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. 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 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 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. Rewrite the integrand:

      3. Integrate term-by-term:

        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:

        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:

        1. The integral of sine is negative cosine:

        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. Let .

        Then let and substitute :

        1. Integrate term-by-term:

          1. The integral of is when :

          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. Let .

        Then let and substitute :

        1. Integrate term-by-term:

          1. The integral of is when :

          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. Let .

        Then let and substitute :

        1. Integrate term-by-term:

          1. The integral of is when :

          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:

    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:

    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:

    The result is:

  3. Now simplify:

  4. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                                     9            3   
 |                                           5            7      128*cos (x)   20*cos (x)
 | sin(5*x)*cos(4*x) dx = C - cos(x) - 24*cos (x) + 32*cos (x) - ----------- + ----------
 |                                                                    9            3     
/                                                                                        
$$-{{\cos \left(9\,x\right)}\over{18}}-{{\cos x}\over{2}}$$
The graph
The answer [src]
5   5*cos(4)*cos(5)   4*sin(4)*sin(5)
- - --------------- - ---------------
9          9                 9       
$${{5}\over{9}}-{{\cos 9+9\,\cos 1}\over{18}}$$
=
=
5   5*cos(4)*cos(5)   4*sin(4)*sin(5)
- - --------------- - ---------------
9          9                 9       
$$- \frac{4 \sin{\left(4 \right)} \sin{\left(5 \right)}}{9} - \frac{5 \cos{\left(4 \right)} \cos{\left(5 \right)}}{9} + \frac{5}{9}$$
Numerical answer [src]
0.336022750503968
0.336022750503968
The graph
Integral of sin(5x)*cos(4x) dx

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