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x^5*cos(x)

Integral of x^5*cos(x) dx

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The solution

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01x5cos(x)dx\int\limits_{0}^{1} x^{5} \cos{\left(x \right)}\, dx
Integral(x^5*cos(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)=x5u{\left(x \right)} = x^{5} and let dv(x)=cos(x)\operatorname{dv}{\left(x \right)} = \cos{\left(x \right)}.

    Then du(x)=5x4\operatorname{du}{\left(x \right)} = 5 x^{4}.

    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.

  2. Use integration by parts:

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

    Let u(x)=5x4u{\left(x \right)} = 5 x^{4} and let dv(x)=sin(x)\operatorname{dv}{\left(x \right)} = \sin{\left(x \right)}.

    Then du(x)=20x3\operatorname{du}{\left(x \right)} = 20 x^{3}.

    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.

  3. Use integration by parts:

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

    Let u(x)=20x3u{\left(x \right)} = - 20 x^{3} and let dv(x)=cos(x)\operatorname{dv}{\left(x \right)} = \cos{\left(x \right)}.

    Then du(x)=60x2\operatorname{du}{\left(x \right)} = - 60 x^{2}.

    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.

  4. Use integration by parts:

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

    Let u(x)=60x2u{\left(x \right)} = - 60 x^{2} and let dv(x)=sin(x)\operatorname{dv}{\left(x \right)} = \sin{\left(x \right)}.

    Then du(x)=120x\operatorname{du}{\left(x \right)} = - 120 x.

    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.

  5. Use integration by parts:

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

    Let u(x)=120xu{\left(x \right)} = 120 x and let dv(x)=cos(x)\operatorname{dv}{\left(x \right)} = \cos{\left(x \right)}.

    Then du(x)=120\operatorname{du}{\left(x \right)} = 120.

    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.

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

    120sin(x)dx=120sin(x)dx\int 120 \sin{\left(x \right)}\, dx = 120 \int \sin{\left(x \right)}\, dx

    1. The integral of sine is negative cosine:

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

    So, the result is: 120cos(x)- 120 \cos{\left(x \right)}

  7. Add the constant of integration:

    x5sin(x)+5x4cos(x)20x3sin(x)60x2cos(x)+120xsin(x)+120cos(x)+constantx^{5} \sin{\left(x \right)} + 5 x^{4} \cos{\left(x \right)} - 20 x^{3} \sin{\left(x \right)} - 60 x^{2} \cos{\left(x \right)} + 120 x \sin{\left(x \right)} + 120 \cos{\left(x \right)}+ \mathrm{constant}


The answer is:

x5sin(x)+5x4cos(x)20x3sin(x)60x2cos(x)+120xsin(x)+120cos(x)+constantx^{5} \sin{\left(x \right)} + 5 x^{4} \cos{\left(x \right)} - 20 x^{3} \sin{\left(x \right)} - 60 x^{2} \cos{\left(x \right)} + 120 x \sin{\left(x \right)} + 120 \cos{\left(x \right)}+ \mathrm{constant}

The answer (Indefinite) [src]
  /                                                                                                    
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 |  5                               5              2              3             4                      
 | x *cos(x) dx = C + 120*cos(x) + x *sin(x) - 60*x *cos(x) - 20*x *sin(x) + 5*x *cos(x) + 120*x*sin(x)
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x5cos(x)dx=C+x5sin(x)+5x4cos(x)20x3sin(x)60x2cos(x)+120xsin(x)+120cos(x)\int x^{5} \cos{\left(x \right)}\, dx = C + x^{5} \sin{\left(x \right)} + 5 x^{4} \cos{\left(x \right)} - 20 x^{3} \sin{\left(x \right)} - 60 x^{2} \cos{\left(x \right)} + 120 x \sin{\left(x \right)} + 120 \cos{\left(x \right)}
The graph
0.001.000.100.200.300.400.500.600.700.800.900200
The answer [src]
-120 + 65*cos(1) + 101*sin(1)
120+65cos(1)+101sin(1)-120 + 65 \cos{\left(1 \right)} + 101 \sin{\left(1 \right)}
=
=
-120 + 65*cos(1) + 101*sin(1)
120+65cos(1)+101sin(1)-120 + 65 \cos{\left(1 \right)} + 101 \sin{\left(1 \right)}
-120 + 65*cos(1) + 101*sin(1)
Numerical answer [src]
0.108219347026629
0.108219347026629
The graph
Integral of x^5*cos(x) dx

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