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Integral of (8x-1)sin(5x+3) dx

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The solution

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 |  (8*x - 1)*sin(5*x + 3) dx
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$$\int\limits_{0}^{1} \left(8 x - 1\right) \sin{\left(5 x + 3 \right)}\, dx$$
Integral((8*x - 1)*sin(5*x + 3), (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(3 + 5*x)   8*sin(3 + 5*x)   8*x*cos(3 + 5*x)
 | (8*x - 1)*sin(5*x + 3) dx = C + ------------ + -------------- - ----------------
 |                                      5               25                5        
/                                                                                  
$$\int \left(8 x - 1\right) \sin{\left(5 x + 3 \right)}\, dx = C - \frac{8 x \cos{\left(5 x + 3 \right)}}{5} + \frac{8 \sin{\left(5 x + 3 \right)}}{25} + \frac{\cos{\left(5 x + 3 \right)}}{5}$$
The graph
The answer [src]
  8*sin(3)   7*cos(8)   cos(3)   8*sin(8)
- -------- - -------- - ------ + --------
     25         5         5         25   
$$- \frac{8 \sin{\left(3 \right)}}{25} - \frac{\cos{\left(3 \right)}}{5} - \frac{7 \cos{\left(8 \right)}}{5} + \frac{8 \sin{\left(8 \right)}}{25}$$
=
=
  8*sin(3)   7*cos(8)   cos(3)   8*sin(8)
- -------- - -------- - ------ + --------
     25         5         5         25   
$$- \frac{8 \sin{\left(3 \right)}}{25} - \frac{\cos{\left(3 \right)}}{5} - \frac{7 \cos{\left(8 \right)}}{5} + \frac{8 \sin{\left(8 \right)}}{25}$$
-8*sin(3)/25 - 7*cos(8)/5 - cos(3)/5 + 8*sin(8)/25
Numerical answer [src]
0.673134782992473
0.673134782992473

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