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x^2*sin3x*dx

Integral of x^2*sin3x*dx dx

Limits of integration:

from to
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The graph:

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Piecewise:

The solution

You have entered [src]
  1               
  /               
 |                
 |   2            
 |  x *sin(3*x) dx
 |                
/                 
0                 
$$\int\limits_{0}^{1} x^{2} \sin{\left(3 x \right)}\, dx$$
Integral(x^2*sin(3*x), (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                   2*cos(3*x)   x *cos(3*x)   2*x*sin(3*x)
 | x *sin(3*x) dx = C + ---------- - ----------- + ------------
 |                          27            3             9      
/                                                              
$$\int x^{2} \sin{\left(3 x \right)}\, dx = C - \frac{x^{2} \cos{\left(3 x \right)}}{3} + \frac{2 x \sin{\left(3 x \right)}}{9} + \frac{2 \cos{\left(3 x \right)}}{27}$$
The graph
The answer [src]
  2    7*cos(3)   2*sin(3)
- -- - -------- + --------
  27      27         9    
$$- \frac{2}{27} + \frac{2 \sin{\left(3 \right)}}{9} - \frac{7 \cos{\left(3 \right)}}{27}$$
=
=
  2    7*cos(3)   2*sin(3)
- -- - -------- + --------
  27      27         9    
$$- \frac{2}{27} + \frac{2 \sin{\left(3 \right)}}{9} - \frac{7 \cos{\left(3 \right)}}{27}$$
-2/27 - 7*cos(3)/27 + 2*sin(3)/9
Numerical answer [src]
0.213950649057864
0.213950649057864
The graph
Integral of x^2*sin3x*dx dx

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