Mister Exam

Integral of cosxsin3x dx

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The solution

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01sin(3x)cos(x)dx\int\limits_{0}^{1} \sin{\left(3 x \right)} \cos{\left(x \right)}\, dx
Integral(cos(x)*sin(3*x), (x, 0, 1))
Detail solution
  1. Rewrite the integrand:

    sin(3x)cos(x)=4sin3(x)cos(x)+3sin(x)cos(x)\sin{\left(3 x \right)} \cos{\left(x \right)} = - 4 \sin^{3}{\left(x \right)} \cos{\left(x \right)} + 3 \sin{\left(x \right)} \cos{\left(x \right)}

  2. Integrate term-by-term:

    1. The integral of a constant times a function is the constant times the integral of the function:

      (4sin3(x)cos(x))dx=4sin3(x)cos(x)dx\int \left(- 4 \sin^{3}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 4 \int \sin^{3}{\left(x \right)} \cos{\left(x \right)}\, dx

      1. There are multiple ways to do this integral.

        Method #1

        1. Let u=sin(x)u = \sin{\left(x \right)}.

          Then let du=cos(x)dxdu = \cos{\left(x \right)} dx and substitute dudu:

          u3du\int u^{3}\, du

          1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n1n \neq -1:

            u3du=u44\int u^{3}\, du = \frac{u^{4}}{4}

          Now substitute uu back in:

          sin4(x)4\frac{\sin^{4}{\left(x \right)}}{4}

        Method #2

        1. Rewrite the integrand:

          sin3(x)cos(x)=(1cos2(x))sin(x)cos(x)\sin^{3}{\left(x \right)} \cos{\left(x \right)} = \left(1 - \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)}

        2. Let u=cos2(x)u = - \cos^{2}{\left(x \right)}.

          Then let du=2sin(x)cos(x)dxdu = 2 \sin{\left(x \right)} \cos{\left(x \right)} dx and substitute dudu:

          (u2+12)du\int \left(\frac{u}{2} + \frac{1}{2}\right)\, du

          1. Integrate term-by-term:

            1. The integral of a constant times a function is the constant times the integral of the function:

              u2du=udu2\int \frac{u}{2}\, du = \frac{\int u\, du}{2}

              1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n1n \neq -1:

                udu=u22\int u\, du = \frac{u^{2}}{2}

              So, the result is: u24\frac{u^{2}}{4}

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

              12du=u2\int \frac{1}{2}\, du = \frac{u}{2}

            The result is: u24+u2\frac{u^{2}}{4} + \frac{u}{2}

          Now substitute uu back in:

          cos4(x)4cos2(x)2\frac{\cos^{4}{\left(x \right)}}{4} - \frac{\cos^{2}{\left(x \right)}}{2}

      So, the result is: sin4(x)- \sin^{4}{\left(x \right)}

    1. The integral of a constant times a function is the constant times the integral of the function:

      3sin(x)cos(x)dx=3sin(x)cos(x)dx\int 3 \sin{\left(x \right)} \cos{\left(x \right)}\, dx = 3 \int \sin{\left(x \right)} \cos{\left(x \right)}\, dx

      1. Let u=cos(x)u = \cos{\left(x \right)}.

        Then let du=sin(x)dxdu = - \sin{\left(x \right)} dx and substitute du- du:

        (u)du\int \left(- u\right)\, du

        1. The integral of a constant times a function is the constant times the integral of the function:

          udu=udu\int u\, du = - \int u\, du

          1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n1n \neq -1:

            udu=u22\int u\, du = \frac{u^{2}}{2}

          So, the result is: u22- \frac{u^{2}}{2}

        Now substitute uu back in:

        cos2(x)2- \frac{\cos^{2}{\left(x \right)}}{2}

      So, the result is: 3cos2(x)2- \frac{3 \cos^{2}{\left(x \right)}}{2}

    The result is: sin4(x)3cos2(x)2- \sin^{4}{\left(x \right)} - \frac{3 \cos^{2}{\left(x \right)}}{2}

  3. Add the constant of integration:

    sin4(x)3cos2(x)2+constant- \sin^{4}{\left(x \right)} - \frac{3 \cos^{2}{\left(x \right)}}{2}+ \mathrm{constant}


The answer is:

sin4(x)3cos2(x)2+constant- \sin^{4}{\left(x \right)} - \frac{3 \cos^{2}{\left(x \right)}}{2}+ \mathrm{constant}

The answer (Indefinite) [src]
  /                                        2   
 |                             4      3*cos (x)
 | cos(x)*sin(3*x) dx = C - sin (x) - ---------
 |                                        2    
/                                              
sin(3x)cos(x)dx=Csin4(x)3cos2(x)2\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
0.001.000.100.200.300.400.500.600.700.800.901-1
The answer [src]
3   3*cos(1)*cos(3)   sin(1)*sin(3)
- - --------------- - -------------
8          8                8      
sin(1)sin(3)83cos(1)cos(3)8+38- \frac{\sin{\left(1 \right)} \sin{\left(3 \right)}}{8} - \frac{3 \cos{\left(1 \right)} \cos{\left(3 \right)}}{8} + \frac{3}{8}
=
=
3   3*cos(1)*cos(3)   sin(1)*sin(3)
- - --------------- - -------------
8          8                8      
sin(1)sin(3)83cos(1)cos(3)8+38- \frac{\sin{\left(1 \right)} \sin{\left(3 \right)}}{8} - \frac{3 \cos{\left(1 \right)} \cos{\left(3 \right)}}{8} + \frac{3}{8}
3/8 - 3*cos(1)*cos(3)/8 - sin(1)*sin(3)/8
Numerical answer [src]
0.560742161744737
0.560742161744737
The graph
Integral of cosxsin3x dx

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