Mister Exam

Other calculators

Solving integrals/ x^2sin4x

Integral of x^2sin4x dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  0               
  /               
 |                
 |   2            
 |  x *sin(4*x) dx
 |                
/                 
pi                
--                
8
∫π80x2sin⁡(4x) dx\int\limits_{\frac{\pi}{8}}^{0} x^{2} \sin{\left(4 x \right)}\, dx 
Integral(x^2*sin(4*x), (x, pi/8, 0))
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⁡(4x)\operatorname{dv}{\left(x \right)} = \sin{\left(4 x \right)}.

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

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

    1. Let u=4xu = 4 x.

      Then let du=4dxdu = 4 dx and substitute du4\frac{du}{4}:

      ∫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) du=∫sin⁡(u) du4\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{4}

        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)4- \frac{\cos{\left(u \right)}}{4}

      Now substitute uu back in:

      −cos⁡(4x)4- \frac{\cos{\left(4 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⁡(4x)\operatorname{dv}{\left(x \right)} = \cos{\left(4 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=4xu = 4 x.

      Then let du=4dxdu = 4 dx and substitute du4\frac{du}{4}:

      ∫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) du=∫cos⁡(u) du4\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{4}

        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)4\frac{\sin{\left(u \right)}}{4}

      Now substitute uu back in:

      sin⁡(4x)4\frac{\sin{\left(4 x \right)}}{4}

    Now evaluate the sub-integral.

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

    ∫(−sin⁡(4x)8) dx=−∫sin⁡(4x) dx8\int \left(- \frac{\sin{\left(4 x \right)}}{8}\right)\, dx = - \frac{\int \sin{\left(4 x \right)}\, dx}{8}

    1. Let u=4xu = 4 x.

      Then let du=4dxdu = 4 dx and substitute du4\frac{du}{4}:

      ∫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) du=∫sin⁡(u) du4\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{4}

        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)4- \frac{\cos{\left(u \right)}}{4}

      Now substitute uu back in:

      −cos⁡(4x)4- \frac{\cos{\left(4 x \right)}}{4}

    So, the result is: cos⁡(4x)32\frac{\cos{\left(4 x \right)}}{32}

  4. Add the constant of integration:

    −x2cos⁡(4x)4+xsin⁡(4x)8+cos⁡(4x)32+constant- \frac{x^{2} \cos{\left(4 x \right)}}{4} + \frac{x \sin{\left(4 x \right)}}{8} + \frac{\cos{\left(4 x \right)}}{32}+ \mathrm{constant}

The answer is:

−x2cos⁡(4x)4+xsin⁡(4x)8+cos⁡(4x)32+constant- \frac{x^{2} \cos{\left(4 x \right)}}{4} + \frac{x \sin{\left(4 x \right)}}{8} + \frac{\cos{\left(4 x \right)}}{32}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                                        
 |                                  2                      
 |  2                   cos(4*x)   x *cos(4*x)   x*sin(4*x)
 | x *sin(4*x) dx = C + -------- - ----------- + ----------
 |                         32           4            8     
/
∫x2sin⁡(4x) dx=C−x2cos⁡(4x)4+xsin⁡(4x)8+cos⁡(4x)32\int x^{2} \sin{\left(4 x \right)}\, dx = C - \frac{x^{2} \cos{\left(4 x \right)}}{4} + \frac{x \sin{\left(4 x \right)}}{8} + \frac{\cos{\left(4 x \right)}}{32}
Simplify
The graph
0.000.050.100.150.200.250.300.350.00.2
Plot the graph f(x)
The answer [src] 
1    pi
-- - --
32   64
132−π64\frac{1}{32} - \frac{\pi}{64} 
=
= 
1    pi
-- - --
32   64
132−π64\frac{1}{32} - \frac{\pi}{64} 
1/32 - pi/64
Numerical answer [src] 
-0.0178373852123405
-0.0178373852123405

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

    © Mister Exam – Calculator