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x^3sinx
Solving integrals/ x^3sinx

Integral of x^3sinx dx

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The solution

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∫01x3sin⁡(x) dx\int\limits_{0}^{1} x^{3} \sin{\left(x \right)}\, dx 
Integral(x^3*sin(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)=x3u{\left(x \right)} = x^{3} and let dv⁡(x)=sin⁡(x)\operatorname{dv}{\left(x \right)} = \sin{\left(x \right)}.

    Then du⁡(x)=3x2\operatorname{du}{\left(x \right)} = 3 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)=−3x2u{\left(x \right)} = - 3 x^{2} and let dv⁡(x)=cos⁡(x)\operatorname{dv}{\left(x \right)} = \cos{\left(x \right)}.

    Then du⁡(x)=−6x\operatorname{du}{\left(x \right)} = - 6 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)=−6xu{\left(x \right)} = - 6 x and let dv⁡(x)=sin⁡(x)\operatorname{dv}{\left(x \right)} = \sin{\left(x \right)}.

    Then du⁡(x)=−6\operatorname{du}{\left(x \right)} = -6.

    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:

    ∫6cos⁡(x) dx=6∫cos⁡(x) dx\int 6 \cos{\left(x \right)}\, dx = 6 \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: 6sin⁡(x)6 \sin{\left(x \right)}

  5. Add the constant of integration:

    −x3cos⁡(x)+3x2sin⁡(x)+6xcos⁡(x)−6sin⁡(x)+constant- x^{3} \cos{\left(x \right)} + 3 x^{2} \sin{\left(x \right)} + 6 x \cos{\left(x \right)} - 6 \sin{\left(x \right)}+ \mathrm{constant}

The answer is:

−x3cos⁡(x)+3x2sin⁡(x)+6xcos⁡(x)−6sin⁡(x)+constant- x^{3} \cos{\left(x \right)} + 3 x^{2} \sin{\left(x \right)} + 6 x \cos{\left(x \right)} - 6 \sin{\left(x \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                                                  
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 | x *sin(x) dx = C - 6*sin(x) - x *cos(x) + 3*x *sin(x) + 6*x*cos(x)
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∫x3sin⁡(x) dx=C−x3cos⁡(x)+3x2sin⁡(x)+6xcos⁡(x)−6sin⁡(x)\int x^{3} \sin{\left(x \right)}\, dx = C - x^{3} \cos{\left(x \right)} + 3 x^{2} \sin{\left(x \right)} + 6 x \cos{\left(x \right)} - 6 \sin{\left(x \right)}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.900.01.0
Plot the graph f(x)
The answer [src] 
-3*sin(1) + 5*cos(1)
−3sin⁡(1)+5cos⁡(1)- 3 \sin{\left(1 \right)} + 5 \cos{\left(1 \right)} 
=
= 
-3*sin(1) + 5*cos(1)
−3sin⁡(1)+5cos⁡(1)- 3 \sin{\left(1 \right)} + 5 \cos{\left(1 \right)} 
-3*sin(1) + 5*cos(1)
Numerical answer [src] 
0.177098574917009
0.177098574917009
The graph
Integral of x^3sinx dx

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

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