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Integral of (3x+2)sin4xdx dx

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The solution

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  1                      
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 |  (3*x + 2)*sin(4*x) dx
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$$\int\limits_{0}^{1} \left(3 x + 2\right) \sin{\left(4 x \right)}\, dx$$
Integral((3*x + 2)*sin(4*x), (x, 0, 1))
Detail solution
  1. There are multiple ways to do this integral.

    Method #1

    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 #2

    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 #3

    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]
  /                                                                
 |                             cos(4*x)   3*sin(4*x)   3*x*cos(4*x)
 | (3*x + 2)*sin(4*x) dx = C - -------- + ---------- - ------------
 |                                2           16            4      
/                                                                  
$$\int \left(3 x + 2\right) \sin{\left(4 x \right)}\, dx = C - \frac{3 x \cos{\left(4 x \right)}}{4} + \frac{3 \sin{\left(4 x \right)}}{16} - \frac{\cos{\left(4 x \right)}}{2}$$
The graph
The answer [src]
1   5*cos(4)   3*sin(4)
- - -------- + --------
2      4          16   
$$\frac{3 \sin{\left(4 \right)}}{16} + \frac{1}{2} - \frac{5 \cos{\left(4 \right)}}{4}$$
=
=
1   5*cos(4)   3*sin(4)
- - -------- + --------
2      4          16   
$$\frac{3 \sin{\left(4 \right)}}{16} + \frac{1}{2} - \frac{5 \cos{\left(4 \right)}}{4}$$
1/2 - 5*cos(4)/4 + 3*sin(4)/16
Numerical answer [src]
1.17515405820928
1.17515405820928

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