Mister Exam

Other calculators

x^2sin(x)^2
Solving integrals/ x^2sin(x)^2

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

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  1              
  /              
 |               
 |   2    2      
 |  x *sin (x) dx
 |               
/                
0
∫01x2sin⁡2(x) dx\int\limits_{0}^{1} x^{2} \sin^{2}{\left(x \right)}\, dx 
Integral(x^2*sin(x)^2, (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)=sin⁡2(x)\operatorname{dv}{\left(x \right)} = \sin^{2}{\left(x \right)}.

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

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

    1. Rewrite the integrand:

      sin⁡2(x)=12−cos⁡(2x)2\sin^{2}{\left(x \right)} = \frac{1}{2} - \frac{\cos{\left(2 x \right)}}{2}

    2. Integrate term-by-term:

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

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

      1. 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)4 du\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)2 du=∫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}

      The result is: x2−sin⁡(2x)4\frac{x}{2} - \frac{\sin{\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)=2x−sin⁡(2x)\operatorname{dv}{\left(x \right)} = 2 x - \sin{\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. Integrate term-by-term:

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

        ∫2x dx=2∫x dx\int 2 x\, dx = 2 \int x\, dx

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

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

        So, the result is: x2x^{2}

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

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

        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)4 du\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)2 du=∫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=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 u\, du

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

                  ∫(−u) du=−∫u du\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 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)}

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

      The result is: x2+cos⁡(2x)2x^{2} + \frac{\cos{\left(2 x \right)}}{2}

    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:

      ∫x22 dx=∫x2 dx2\int \frac{x^{2}}{2}\, dx = \frac{\int x^{2}\, dx}{2}

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

        ∫x2 dx=x33\int x^{2}\, dx = \frac{x^{3}}{3}

      So, the result is: x36\frac{x^{3}}{6}

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

      ∫cos⁡(2x)4 dx=∫cos⁡(2x) dx4\int \frac{\cos{\left(2 x \right)}}{4}\, dx = \frac{\int \cos{\left(2 x \right)}\, dx}{4}

      1. Let u=2xu = 2 x.

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

        ∫cos⁡(u)4 du\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)2 du=∫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)8\frac{\sin{\left(2 x \right)}}{8}

    The result is: x36+sin⁡(2x)8\frac{x^{3}}{6} + \frac{\sin{\left(2 x \right)}}{8}

  4. Now simplify:

    x36−x2sin⁡(2x)4−xcos⁡(2x)4+sin⁡(2x)8\frac{x^{3}}{6} - \frac{x^{2} \sin{\left(2 x \right)}}{4} - \frac{x \cos{\left(2 x \right)}}{4} + \frac{\sin{\left(2 x \right)}}{8}

  5. Add the constant of integration:

    x36−x2sin⁡(2x)4−xcos⁡(2x)4+sin⁡(2x)8+constant\frac{x^{3}}{6} - \frac{x^{2} \sin{\left(2 x \right)}}{4} - \frac{x \cos{\left(2 x \right)}}{4} + \frac{\sin{\left(2 x \right)}}{8}+ \mathrm{constant}

The answer is:

x36−x2sin⁡(2x)4−xcos⁡(2x)4+sin⁡(2x)8+constant\frac{x^{3}}{6} - \frac{x^{2} \sin{\left(2 x \right)}}{4} - \frac{x \cos{\left(2 x \right)}}{4} + \frac{\sin{\left(2 x \right)}}{8}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                                          / 2   cos(2*x)\
 |                      3                                  x*|x  + --------|
 |  2    2             x    sin(2*x)    2 /x   sin(2*x)\     \        2    /
 | x *sin (x) dx = C + -- + -------- + x *|- - --------| - -----------------
 |                     6       8          \2      4    /           2        
/
−(6 x2−3) sin⁡(2 x)+6 x cos⁡(2 x)−4 x324-{{\left(6\,x^2-3\right)\,\sin \left(2\,x\right)+6\,x\,\cos \left(2 \,x\right)-4\,x^3}\over{24}}
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] 
     2           2                   
  cos (1)   5*sin (1)   cos(1)*sin(1)
- ------- + --------- - -------------
     12         12            4
−3 sin⁡2+6 cos⁡2−424-{{3\,\sin 2+6\,\cos 2-4}\over{24}} 
=
= 
     2           2                   
  cos (1)   5*sin (1)   cos(1)*sin(1)
- ------- + --------- - -------------
     12         12            4
−sin⁡(1)cos⁡(1)4−cos⁡2(1)12+5sin⁡2(1)12- \frac{\sin{\left(1 \right)} \cos{\left(1 \right)}}{4} - \frac{\cos^{2}{\left(1 \right)}}{12} + \frac{5 \sin^{2}{\left(1 \right)}}{12}
Simplify
Numerical answer [src] 
0.157041197450242
0.157041197450242
The graph
Integral of x^2sin(x)^2 dx

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

    © Mister Exam – Calculator