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

Integral of x^2*cos(2*x) dx

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 -pi              
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 |  x *cos(2*x) dx
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pi
∫π−πx2cos⁡(2x) dx\int\limits_{\pi}^{- \pi} x^{2} \cos{\left(2 x \right)}\, dx 
Integral(x^2*cos(2*x), (x, pi, -pi))
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⁡(2x)\operatorname{dv}{\left(x \right)} = \cos{\left(2 x \right)}.

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

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

    1. Let u=2xu = 2 x.

      Then let du=2dxdu = 2 dx and substitute du2\frac{du}{2}:

      ∫cos⁡(u)2 du\int \frac{\cos{\left(u \right)}}{2}\, du

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

        ∫cos⁡(u) du=∫cos⁡(u) du2\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{2}

        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)2\frac{\sin{\left(u \right)}}{2}

      Now substitute uu back in:

      sin⁡(2x)2\frac{\sin{\left(2 x \right)}}{2}

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

    Then du⁡(x)=1\operatorname{du}{\left(x \right)} = 1.

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

    1. There are multiple ways to do this integral.

      Method #1

      1. Let u=2xu = 2 x.

        Then let du=2dxdu = 2 dx and substitute du2\frac{du}{2}:

        ∫sin⁡(u)2 du\int \frac{\sin{\left(u \right)}}{2}\, du

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

          ∫sin⁡(u) du=∫sin⁡(u) du2\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{2}

          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)2- \frac{\cos{\left(u \right)}}{2}

        Now substitute uu back in:

        −cos⁡(2x)2- \frac{\cos{\left(2 x \right)}}{2}

      Method #2

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

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

        1. There are multiple ways to do this integral.

          Method #1

          1. Let u=cos⁡(x)u = \cos{\left(x \right)}.

            Then let du=−sin⁡(x)dxdu = - \sin{\left(x \right)} dx and substitute −du- du:

            ∫(−u) du\int \left(- u\right)\, du

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

              ∫u du=−∫u du\int u\, du = - \int u\, du

              1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

                ∫u du=u22\int u\, du = \frac{u^{2}}{2}

              So, the result is: −u22- \frac{u^{2}}{2}

            Now substitute uu back in:

            −cos⁡2(x)2- \frac{\cos^{2}{\left(x \right)}}{2}

          Method #2

          1. Let u=sin⁡(x)u = \sin{\left(x \right)}.

            Then let du=cos⁡(x)dxdu = \cos{\left(x \right)} dx and substitute dudu:

            ∫u du\int u\, du

            1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

              ∫u du=u22\int u\, du = \frac{u^{2}}{2}

            Now substitute uu back in:

            sin⁡2(x)2\frac{\sin^{2}{\left(x \right)}}{2}

        So, the result is: −cos⁡2(x)- \cos^{2}{\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:

    ∫(−cos⁡(2x)2) dx=−∫cos⁡(2x) dx2\int \left(- \frac{\cos{\left(2 x \right)}}{2}\right)\, dx = - \frac{\int \cos{\left(2 x \right)}\, dx}{2}

    1. Let u=2xu = 2 x.

      Then let du=2dxdu = 2 dx and substitute du2\frac{du}{2}:

      ∫cos⁡(u)2 du\int \frac{\cos{\left(u \right)}}{2}\, du

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

        ∫cos⁡(u) du=∫cos⁡(u) du2\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{2}

        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)2\frac{\sin{\left(u \right)}}{2}

      Now substitute uu back in:

      sin⁡(2x)2\frac{\sin{\left(2 x \right)}}{2}

    So, the result is: −sin⁡(2x)4- \frac{\sin{\left(2 x \right)}}{4}

  4. Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src] 
  /                                                        
 |                                               2         
 |  2                   sin(2*x)   x*cos(2*x)   x *sin(2*x)
 | x *cos(2*x) dx = C - -------- + ---------- + -----------
 |                         4           2             2     
/
∫x2cos⁡(2x) dx=C+x2sin⁡(2x)2+xcos⁡(2x)2−sin⁡(2x)4\int x^{2} \cos{\left(2 x \right)}\, dx = C + \frac{x^{2} \sin{\left(2 x \right)}}{2} + \frac{x \cos{\left(2 x \right)}}{2} - \frac{\sin{\left(2 x \right)}}{4}
Simplify
The graph
-3.0-2.5-2.0-1.5-1.0-0.50.00.51.01.52.02.53.020-10
Plot the graph f(x)
The answer [src] 
-pi
−π- \pi 
=
= 
-pi
−π- \pi 
-pi
Numerical answer [src] 
-3.14159265358979
-3.14159265358979

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

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