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4x^3*sin(x)
Solving integrals/ 4x^3*sin(x)

Integral of 4x^3*sin(x) dx

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∫124x3sin⁡(x) dx\int\limits_{1}^{2} 4 x^{3} \sin{\left(x \right)}\, dx 
Integral((4*x^3)*sin(x), (x, 1, 2))
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)=4x3u{\left(x \right)} = 4 x^{3} and let dv⁡(x)=sin⁡(x)\operatorname{dv}{\left(x \right)} = \sin{\left(x \right)}.

    Then du⁡(x)=12x2\operatorname{du}{\left(x \right)} = 12 x^{2}.

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

    1. The integral of sine is negative cosine:

      ∫sin⁡(x) dx=−cos⁡(x)\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}

    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)=−12x2u{\left(x \right)} = - 12 x^{2} and let dv⁡(x)=cos⁡(x)\operatorname{dv}{\left(x \right)} = \cos{\left(x \right)}.

    Then du⁡(x)=−24x\operatorname{du}{\left(x \right)} = - 24 x.

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

    1. The integral of cosine is sine:

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

    Now evaluate the sub-integral.

  3. 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)=−24xu{\left(x \right)} = - 24 x and let dv⁡(x)=sin⁡(x)\operatorname{dv}{\left(x \right)} = \sin{\left(x \right)}.

    Then du⁡(x)=−24\operatorname{du}{\left(x \right)} = -24.

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

    1. The integral of sine is negative cosine:

      ∫sin⁡(x) dx=−cos⁡(x)\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}

    Now evaluate the sub-integral.

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

    ∫24cos⁡(x) dx=24∫cos⁡(x) dx\int 24 \cos{\left(x \right)}\, dx = 24 \int \cos{\left(x \right)}\, dx

    1. The integral of cosine is sine:

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

    So, the result is: 24sin⁡(x)24 \sin{\left(x \right)}

  5. Add the constant of integration:

    −4x3cos⁡(x)+12x2sin⁡(x)+24xcos⁡(x)−24sin⁡(x)+constant- 4 x^{3} \cos{\left(x \right)} + 12 x^{2} \sin{\left(x \right)} + 24 x \cos{\left(x \right)} - 24 \sin{\left(x \right)}+ \mathrm{constant}

The answer is:

−4x3cos⁡(x)+12x2sin⁡(x)+24xcos⁡(x)−24sin⁡(x)+constant- 4 x^{3} \cos{\left(x \right)} + 12 x^{2} \sin{\left(x \right)} + 24 x \cos{\left(x \right)} - 24 \sin{\left(x \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                                                         
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 |    3                                3              2                     
 | 4*x *sin(x) dx = C - 24*sin(x) - 4*x *cos(x) + 12*x *sin(x) + 24*x*cos(x)
 |                                                                          
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∫4x3sin⁡(x) dx=C−4x3cos⁡(x)+12x2sin⁡(x)+24xcos⁡(x)−24sin⁡(x)\int 4 x^{3} \sin{\left(x \right)}\, dx = C - 4 x^{3} \cos{\left(x \right)} + 12 x^{2} \sin{\left(x \right)} + 24 x \cos{\left(x \right)} - 24 \sin{\left(x \right)}
Simplify
The graph
1.002.001.101.201.301.401.501.601.701.801.90050
Plot the graph f(x)
The answer [src] 
-20*cos(1) + 12*sin(1) + 16*cos(2) + 24*sin(2)
−20cos⁡(1)+16cos⁡(2)+12sin⁡(1)+24sin⁡(2)- 20 \cos{\left(1 \right)} + 16 \cos{\left(2 \right)} + 12 \sin{\left(1 \right)} + 24 \sin{\left(2 \right)} 
=
= 
-20*cos(1) + 12*sin(1) + 16*cos(2) + 24*sin(2)
−20cos⁡(1)+16cos⁡(2)+12sin⁡(1)+24sin⁡(2)- 20 \cos{\left(1 \right)} + 16 \cos{\left(2 \right)} + 12 \sin{\left(1 \right)} + 24 \sin{\left(2 \right)} 
-20*cos(1) + 12*sin(1) + 16*cos(2) + 24*sin(2)
Numerical answer [src] 
14.456394559394
14.456394559394
The graph
Integral of 4x^3*sin(x) dx

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

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