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x^2sin2xdx
Solving integrals/ x^2sin2xdx

Integral of x^2sin2xdx dx

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∫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:

      ∫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)=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)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. 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}

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

    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)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}

      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=2∫x2sin⁡(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:

        ∫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)=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. 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 \frac{\sin{\left(2 x \right)}}{2}\, 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)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}

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

        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)=−x2u{\left(x \right)} = - \frac{x}{2} and let dv⁡(x)=cos⁡(2x)\operatorname{dv}{\left(x \right)} = \cos{\left(2 x \right)}.

        Then du⁡(x)=−12\operatorname{du}{\left(x \right)} = - \frac{1}{2}.

        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.

      3. 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)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}

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

      So, the result is: −x2cos⁡(2x)2+xsin⁡(2x)2+cos⁡(2x)4- \frac{x^{2} \cos{\left(2 x \right)}}{2} + \frac{x \sin{\left(2 x \right)}}{2} + \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     
/
∫x2sin⁡(2x) dx=C−x2cos⁡(2x)2+xsin⁡(2x)2+cos⁡(2x)4\int x^{2} \sin{\left(2 x \right)}\, dx = C - \frac{x^{2} \cos{\left(2 x \right)}}{2} + \frac{x \sin{\left(2 x \right)}}{2} + \frac{\cos{\left(2 x \right)}}{4}
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] 
  1   sin(2)   cos(2)
- - + ------ - ------
  4     2        4
−14−cos⁡(2)4+sin⁡(2)2- \frac{1}{4} - \frac{\cos{\left(2 \right)}}{4} + \frac{\sin{\left(2 \right)}}{2} 
=
= 
  1   sin(2)   cos(2)
- - + ------ - ------
  4     2        4
−14−cos⁡(2)4+sin⁡(2)2- \frac{1}{4} - \frac{\cos{\left(2 \right)}}{4} + \frac{\sin{\left(2 \right)}}{2} 
-1/4 + sin(2)/2 - cos(2)/4
Numerical answer [src] 
0.308685422549626
0.308685422549626
The graph
Integral of x^2sin2xdx dx

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

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