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(x^2)*cos(3x)
Solving integrals/ (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:

    ∫u⁡dv⁡=u⁡v⁡−∫v⁡du⁡\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)3 du\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:

    ∫u⁡dv⁡=u⁡v⁡−∫v⁡du⁡\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)3 du\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=−2∫cos⁡(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)3 du\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)9−2sin⁡(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)9−2sin⁡(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)9−2sin⁡(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}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.901-2
Plot the graph f(x)
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.

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