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(6x^2)cos(3x)

Integral of (6x^2)cos(3x) dx

Limits of integration:

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The graph:

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Piecewise:

The solution

You have entered [src]
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 |  6*x *cos(3*x) dx
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$$\int\limits_{0}^{1} 6 x^{2} \cos{\left(3 x \right)}\, dx$$
Integral(6*x^2*cos(3*x), (x, 0, 1))
Detail solution
  1. The integral of a constant times a function is the constant times the integral of the function:

    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 cosine is sine:

          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 sine is negative cosine:

          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 cosine is sine:

          So, the result is:

        Now substitute back in:

      So, the result is:

    So, the result is:

  2. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                                
 |                                                                 
 |    2                   4*sin(3*x)      2            4*x*cos(3*x)
 | 6*x *cos(3*x) dx = C - ---------- + 2*x *sin(3*x) + ------------
 |                            9                             3      
/                                                                  
$${{2\,\left(\left(9\,x^2-2\right)\,\sin \left(3\,x\right)+6\,x\, \cos \left(3\,x\right)\right)}\over{9}}$$
The graph
The answer [src]
4*cos(3)   14*sin(3)
-------- + ---------
   3           9    
$${{2\,\left(7\,\sin 3+6\,\cos 3\right)}\over{9}}$$
=
=
4*cos(3)   14*sin(3)
-------- + ---------
   3           9    
$$\frac{4 \cos{\left(3 \right)}}{3} + \frac{14 \sin{\left(3 \right)}}{9}$$
Numerical answer [src]
-1.10046998292969
-1.10046998292969
The graph
Integral of (6x^2)cos(3x) dx

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