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x^3sin2xdx

Integral of x^3sin2xdx dx

Limits of integration:

from to
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The graph:

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Piecewise:

The solution

You have entered [src]
  1               
  /               
 |                
 |   3            
 |  x *sin(2*x) dx
 |                
/                 
0                 
$$\int\limits_{0}^{1} x^{3} \sin{\left(2 x \right)}\, dx$$
Integral(x^3*sin(2*x), (x, 0, 1))
Detail solution
  1. There are multiple ways to do this integral.

    Method #1

    1. Use integration by parts:

      Let and let .

      Then .

      To find :

      1. There are multiple ways to do this integral.

        Method #1

        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:

        Method #2

        1. 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 is when :

              So, the result is:

            Now substitute back in:

          So, the result is:

      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. 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.

    4. 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:

    Method #2

    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. 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:

        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. 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.

      4. 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]
  /                                                                            
 |                                    3                              2         
 |  3                   3*sin(2*x)   x *cos(2*x)   3*x*cos(2*x)   3*x *sin(2*x)
 | x *sin(2*x) dx = C - ---------- - ----------- + ------------ + -------------
 |                          8             2             4               4      
/                                                                              
$$\int x^{3} \sin{\left(2 x \right)}\, dx = C - \frac{x^{3} \cos{\left(2 x \right)}}{2} + \frac{3 x^{2} \sin{\left(2 x \right)}}{4} + \frac{3 x \cos{\left(2 x \right)}}{4} - \frac{3 \sin{\left(2 x \right)}}{8}$$
The graph
The answer [src]
cos(2)   3*sin(2)
------ + --------
  4         8    
$$\frac{\cos{\left(2 \right)}}{4} + \frac{3 \sin{\left(2 \right)}}{8}$$
=
=
cos(2)   3*sin(2)
------ + --------
  4         8    
$$\frac{\cos{\left(2 \right)}}{4} + \frac{3 \sin{\left(2 \right)}}{8}$$
cos(2)/4 + 3*sin(2)/8
Numerical answer [src]
0.236949825922845
0.236949825922845
The graph
Integral of x^3sin2xdx dx

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