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

Integral of x^2*cos(x) dx

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The solution

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∫01x2cos⁡(x) dx\int\limits_{0}^{1} x^{2} \cos{\left(x \right)}\, dx 
Integral(x^2*cos(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⁡(x)\operatorname{dv}{\left(x \right)} = \cos{\left(x \right)}.

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

  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)=2xu{\left(x \right)} = 2 x and let dv⁡(x)=sin⁡(x)\operatorname{dv}{\left(x \right)} = \sin{\left(x \right)}.

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

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

    ∫(−2cos⁡(x)) dx=−2∫cos⁡(x) dx\int \left(- 2 \cos{\left(x \right)}\right)\, dx = - 2 \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: −2sin⁡(x)- 2 \sin{\left(x \right)}

  4. Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src] 
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∫x2cos⁡(x) dx=C+x2sin⁡(x)+2xcos⁡(x)−2sin⁡(x)\int x^{2} \cos{\left(x \right)}\, dx = C + x^{2} \sin{\left(x \right)} + 2 x \cos{\left(x \right)} - 2 \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] 
-sin(1) + 2*cos(1)
−sin⁡(1)+2cos⁡(1)- \sin{\left(1 \right)} + 2 \cos{\left(1 \right)} 
=
= 
-sin(1) + 2*cos(1)
−sin⁡(1)+2cos⁡(1)- \sin{\left(1 \right)} + 2 \cos{\left(1 \right)} 
-sin(1) + 2*cos(1)
Numerical answer [src] 
0.239133626928383
0.239133626928383
The graph
Integral of x^2*cos(x) dx

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

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