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(3x+2)*sin3x

Integral of (3x+2)*sin3x dx

Limits of integration:

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The graph:

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Piecewise:

The solution

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

    Method #1

    1. Let .

      Then let and substitute :

      1. Integrate term-by-term:

        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 sine is negative cosine:

            Now evaluate the sub-integral.

          2. 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:

          So, the result is:

        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:

        The result is:

      Now substitute back in:

    Method #2

    1. Rewrite the integrand:

    2. Integrate term-by-term:

      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 sine is negative cosine:

              So, the result is:

            Now substitute back in:

          Now evaluate the sub-integral.

        2. 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:

      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:

      The result is:

    Method #3

    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. 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 #4

    1. Rewrite the integrand:

    2. Integrate term-by-term:

      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 sine is negative cosine:

              So, the result is:

            Now substitute back in:

          Now evaluate the sub-integral.

        2. 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:

      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:

      The result is:

  2. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                              
 |                             2*cos(3*x)   sin(3*x)             
 | (3*x + 2)*sin(3*x) dx = C - ---------- + -------- - x*cos(3*x)
 |                                 3           3                 
/                                                                
$${{\sin \left(3\,x\right)-3\,x\,\cos \left(3\,x\right)-2\,\cos \left(3\,x\right)}\over{3}}$$
The graph
The answer [src]
2   5*cos(3)   sin(3)
- - -------- + ------
3      3         3   
$${{\sin 3-5\,\cos 3}\over{3}}+{{2}\over{3}}$$
=
=
2   5*cos(3)   sin(3)
- - -------- + ------
3      3         3   
$$\frac{\sin{\left(3 \right)}}{3} + \frac{2}{3} - \frac{5 \cos{\left(3 \right)}}{3}$$
Numerical answer [src]
2.36369416368736
2.36369416368736
The graph
Integral of (3x+2)*sin3x dx

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