Mister Exam

Other calculators


x^2*sin(2x)

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

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1               
  /               
 |                
 |   2            
 |  x *sin(2*x) dx
 |                
/                 
0                 
01x2sin(2x)dx\int\limits_{0}^{1} x^{2} \sin{\left(2 x \right)}\, dx
Integral(x^2*sin(2*x), (x, 0, 1))
Detail solution
  1. There are multiple ways to do this integral.

    Method #1

    1. Use integration by parts:

      udv=uvvdu\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)=sin(2x)\operatorname{dv}{\left(x \right)} = \sin{\left(2 x \right)}.

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

      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)4du\int \frac{\sin{\left(u \right)}}{4}\, du

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

            sin(u)2du=sin(u)du2\int \frac{\sin{\left(u \right)}}{2}\, 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=2sin(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. Let u=cos(x)u = \cos{\left(x \right)}.

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

            udu\int u\, du

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

              (u)du=udu\int \left(- u\right)\, du = - \int u\, du

              1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n1n \neq -1:

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

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

            Now substitute uu back in:

            cos2(x)2- \frac{\cos^{2}{\left(x \right)}}{2}

          So, the result is: cos2(x)- \cos^{2}{\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)=xu{\left(x \right)} = - x and let dv(x)=cos(2x)\operatorname{dv}{\left(x \right)} = \cos{\left(2 x \right)}.

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

      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)4du\int \frac{\cos{\left(u \right)}}{4}\, du

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

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

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

      (sin(2x)2)dx=sin(2x)dx2\int \left(- \frac{\sin{\left(2 x \right)}}{2}\right)\, dx = - \frac{\int \sin{\left(2 x \right)}\, dx}{2}

      1. Let u=2xu = 2 x.

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

        sin(u)4du\int \frac{\sin{\left(u \right)}}{4}\, du

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

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

      So, the result is: cos(2x)4\frac{\cos{\left(2 x \right)}}{4}

    Method #2

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

      2x2sin(x)cos(x)dx=2x2sin(x)cos(x)dx\int 2 x^{2} \sin{\left(x \right)} \cos{\left(x \right)}\, dx = 2 \int x^{2} \sin{\left(x \right)} \cos{\left(x \right)}\, dx

      1. Use integration by parts:

        udv=uvvdu\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)=sin(x)cos(x)\operatorname{dv}{\left(x \right)} = \sin{\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. Let u=cos(x)u = \cos{\left(x \right)}.

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

          udu\int u\, du

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

            (u)du=udu\int \left(- u\right)\, du = - \int u\, du

            1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n1n \neq -1:

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

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

          Now substitute uu back in:

          cos2(x)2- \frac{\cos^{2}{\left(x \right)}}{2}

        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)=xu{\left(x \right)} = - x and let dv(x)=cos2(x)\operatorname{dv}{\left(x \right)} = \cos^{2}{\left(x \right)}.

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

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

        1. Rewrite the integrand:

          cos2(x)=cos(2x)2+12\cos^{2}{\left(x \right)} = \frac{\cos{\left(2 x \right)}}{2} + \frac{1}{2}

        2. Integrate term-by-term:

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

            cos(2x)2dx=cos(2x)dx2\int \frac{\cos{\left(2 x \right)}}{2}\, 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)4du\int \frac{\cos{\left(u \right)}}{4}\, du

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

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

          1. The integral of a constant is the constant times the variable of integration:

            12dx=x2\int \frac{1}{2}\, dx = \frac{x}{2}

          The result is: x2+sin(2x)4\frac{x}{2} + \frac{\sin{\left(2 x \right)}}{4}

        Now evaluate the sub-integral.

      3. Integrate term-by-term:

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

          (x2)dx=xdx2\int \left(- \frac{x}{2}\right)\, dx = - \frac{\int x\, dx}{2}

          1. The integral of xnx^{n} is xn+1n+1\frac{x^{n + 1}}{n + 1} when n1n \neq -1:

            xdx=x22\int x\, dx = \frac{x^{2}}{2}

          So, the result is: x24- \frac{x^{2}}{4}

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

          (sin(2x)4)dx=sin(2x)dx4\int \left(- \frac{\sin{\left(2 x \right)}}{4}\right)\, dx = - \frac{\int \sin{\left(2 x \right)}\, dx}{4}

          1. Let u=2xu = 2 x.

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

            sin(u)4du\int \frac{\sin{\left(u \right)}}{4}\, du

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

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

          So, the result is: cos(2x)8\frac{\cos{\left(2 x \right)}}{8}

        The result is: x24+cos(2x)8- \frac{x^{2}}{4} + \frac{\cos{\left(2 x \right)}}{8}

      So, the result is: x2cos2(x)x22+2x(x2+sin(2x)4)+cos(2x)4- x^{2} \cos^{2}{\left(x \right)} - \frac{x^{2}}{2} + 2 x \left(\frac{x}{2} + \frac{\sin{\left(2 x \right)}}{4}\right) + \frac{\cos{\left(2 x \right)}}{4}

  2. Add the constant of integration:

    x2cos(2x)2+xsin(2x)2+cos(2x)4+constant- \frac{x^{2} \cos{\left(2 x \right)}}{2} + \frac{x \sin{\left(2 x \right)}}{2} + \frac{\cos{\left(2 x \right)}}{4}+ \mathrm{constant}


The answer is:

x2cos(2x)2+xsin(2x)2+cos(2x)4+constant- \frac{x^{2} \cos{\left(2 x \right)}}{2} + \frac{x \sin{\left(2 x \right)}}{2} + \frac{\cos{\left(2 x \right)}}{4}+ \mathrm{constant}

The answer (Indefinite) [src]
  /                                                        
 |                                               2         
 |  2                   cos(2*x)   x*sin(2*x)   x *cos(2*x)
 | x *sin(2*x) dx = C + -------- + ---------- - -----------
 |                         4           2             2     
/                                                          
4xsin(2x)+(24x2)cos(2x)8{{4\,x\,\sin \left(2\,x\right)+\left(2-4\,x^2\right)\,\cos \left(2 \,x\right)}\over{8}}
The graph
0.001.000.100.200.300.400.500.600.700.800.900.01.0
The answer [src]
  1   sin(2)   cos(2)
- - + ------ - ------
  4     2        4   
2sin2cos2414{{2\,\sin 2-\cos 2}\over{4}}-{{1}\over{4}}
=
=
  1   sin(2)   cos(2)
- - + ------ - ------
  4     2        4   
14cos(2)4+sin(2)2- \frac{1}{4} - \frac{\cos{\left(2 \right)}}{4} + \frac{\sin{\left(2 \right)}}{2}
Numerical answer [src]
0.308685422549626
0.308685422549626
The graph
Integral of x^2*sin(2x) dx

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