Mister Exam

Other calculators


(x^2)sin4xdx

Integral of (x^2)sin4xdx dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1                 
  /                 
 |                  
 |   2              
 |  x *sin(4*x)*1 dx
 |                  
/                   
0                   
$$\int\limits_{0}^{1} x^{2} \sin{\left(4 x \right)} 1\, dx$$
Integral(x^2*sin(4*x)*1, (x, 0, 1))
Detail solution
  1. Use integration by parts:

    Let and let .

    Then .

    To find :

    1. Let .

      Then let and substitute :

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

        1. The integral of sine is negative cosine:

        So, the result is:

      Now substitute back in:

    Now evaluate the sub-integral.

  2. Use integration by parts:

    Let and let .

    Then .

    To find :

    1. Let .

      Then let and substitute :

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

        1. The integral of cosine is sine:

        So, the result is:

      Now substitute back in:

    Now evaluate the sub-integral.

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

    1. Let .

      Then let and substitute :

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

        1. The integral of sine is negative cosine:

        So, the result is:

      Now substitute back in:

    So, the result is:

  4. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                          
 |                                    2                      
 |  2                     cos(4*x)   x *cos(4*x)   x*sin(4*x)
 | x *sin(4*x)*1 dx = C + -------- - ----------- + ----------
 |                           32           4            8     
/                                                            
$${{8\,x\,\sin \left(4\,x\right)+\left(2-16\,x^2\right)\,\cos \left(4 \,x\right)}\over{64}}$$
The graph
The answer [src]
  1    7*cos(4)   sin(4)
- -- - -------- + ------
  32      32        8   
$${{4\,\sin 4-7\,\cos 4}\over{32}}-{{1}\over{32}}$$
=
=
  1    7*cos(4)   sin(4)
- -- - -------- + ------
  32      32        8   
$$\frac{\sin{\left(4 \right)}}{8} - \frac{1}{32} - \frac{7 \cos{\left(4 \right)}}{32}$$
Numerical answer [src]
0.0171342301504241
0.0171342301504241
The graph
Integral of (x^2)sin4xdx dx

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