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(x^2)*cos(3x)

Integral of (x^2)*cos(3x) dx

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01x2cos(3x)dx\int\limits_{0}^{1} x^{2} \cos{\left(3 x \right)}\, dx
Integral(x^2*cos(3*x), (x, 0, 1))
Detail solution
  1. Use integration by parts:

    udv=uvvdu\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}

    Let u(x)=x2u{\left(x \right)} = x^{2} and let dv(x)=cos(3x)\operatorname{dv}{\left(x \right)} = \cos{\left(3 x \right)}.

    Then du(x)=2x\operatorname{du}{\left(x \right)} = 2 x.

    To find v(x)v{\left(x \right)}:

    1. Let u=3xu = 3 x.

      Then let du=3dxdu = 3 dx and substitute du3\frac{du}{3}:

      cos(u)3du\int \frac{\cos{\left(u \right)}}{3}\, du

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

        cos(u)du=cos(u)du3\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{3}

        1. The integral of cosine is sine:

          cos(u)du=sin(u)\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}

        So, the result is: sin(u)3\frac{\sin{\left(u \right)}}{3}

      Now substitute uu back in:

      sin(3x)3\frac{\sin{\left(3 x \right)}}{3}

    Now evaluate the sub-integral.

  2. Use integration by parts:

    udv=uvvdu\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}

    Let u(x)=2x3u{\left(x \right)} = \frac{2 x}{3} and let dv(x)=sin(3x)\operatorname{dv}{\left(x \right)} = \sin{\left(3 x \right)}.

    Then du(x)=23\operatorname{du}{\left(x \right)} = \frac{2}{3}.

    To find v(x)v{\left(x \right)}:

    1. Let u=3xu = 3 x.

      Then let du=3dxdu = 3 dx and substitute du3\frac{du}{3}:

      sin(u)3du\int \frac{\sin{\left(u \right)}}{3}\, du

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

        sin(u)du=sin(u)du3\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{3}

        1. The integral of sine is negative cosine:

          sin(u)du=cos(u)\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}

        So, the result is: cos(u)3- \frac{\cos{\left(u \right)}}{3}

      Now substitute uu back in:

      cos(3x)3- \frac{\cos{\left(3 x \right)}}{3}

    Now evaluate the sub-integral.

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

    (2cos(3x)9)dx=2cos(3x)dx9\int \left(- \frac{2 \cos{\left(3 x \right)}}{9}\right)\, dx = - \frac{2 \int \cos{\left(3 x \right)}\, dx}{9}

    1. Let u=3xu = 3 x.

      Then let du=3dxdu = 3 dx and substitute du3\frac{du}{3}:

      cos(u)3du\int \frac{\cos{\left(u \right)}}{3}\, du

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

        cos(u)du=cos(u)du3\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{3}

        1. The integral of cosine is sine:

          cos(u)du=sin(u)\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}

        So, the result is: sin(u)3\frac{\sin{\left(u \right)}}{3}

      Now substitute uu back in:

      sin(3x)3\frac{\sin{\left(3 x \right)}}{3}

    So, the result is: 2sin(3x)27- \frac{2 \sin{\left(3 x \right)}}{27}

  4. Add the constant of integration:

    x2sin(3x)3+2xcos(3x)92sin(3x)27+constant\frac{x^{2} \sin{\left(3 x \right)}}{3} + \frac{2 x \cos{\left(3 x \right)}}{9} - \frac{2 \sin{\left(3 x \right)}}{27}+ \mathrm{constant}


The answer is:

x2sin(3x)3+2xcos(3x)92sin(3x)27+constant\frac{x^{2} \sin{\left(3 x \right)}}{3} + \frac{2 x \cos{\left(3 x \right)}}{9} - \frac{2 \sin{\left(3 x \right)}}{27}+ \mathrm{constant}

The answer (Indefinite) [src]
  /                                                            
 |                                    2                        
 |  2                   2*sin(3*x)   x *sin(3*x)   2*x*cos(3*x)
 | x *cos(3*x) dx = C - ---------- + ----------- + ------------
 |                          27            3             9      
/                                                              
x2cos(3x)dx=C+x2sin(3x)3+2xcos(3x)92sin(3x)27\int x^{2} \cos{\left(3 x \right)}\, dx = C + \frac{x^{2} \sin{\left(3 x \right)}}{3} + \frac{2 x \cos{\left(3 x \right)}}{9} - \frac{2 \sin{\left(3 x \right)}}{27}
The graph
0.001.000.100.200.300.400.500.600.700.800.901-2
The answer [src]
2*cos(3)   7*sin(3)
-------- + --------
   9          27   
2cos(3)9+7sin(3)27\frac{2 \cos{\left(3 \right)}}{9} + \frac{7 \sin{\left(3 \right)}}{27}
=
=
2*cos(3)   7*sin(3)
-------- + --------
   9          27   
2cos(3)9+7sin(3)27\frac{2 \cos{\left(3 \right)}}{9} + \frac{7 \sin{\left(3 \right)}}{27}
2*cos(3)/9 + 7*sin(3)/27
Numerical answer [src]
-0.183411663821615
-0.183411663821615
The graph
Integral of (x^2)*cos(3x) dx

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