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Integral of (2*x-3)*sin(5*x) dx

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The solution

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 |  (2*x - 3)*sin(5*x) dx
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$$\int\limits_{0}^{1} \left(2 x - 3\right) \sin{\left(5 x \right)}\, dx$$
Integral((2*x - 3)*sin(5*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]
  /                                                                  
 |                             2*sin(5*x)   3*cos(5*x)   2*x*cos(5*x)
 | (2*x - 3)*sin(5*x) dx = C + ---------- + ---------- - ------------
 |                                 25           5             5      
/                                                                    
$$\int \left(2 x - 3\right) \sin{\left(5 x \right)}\, dx = C - \frac{2 x \cos{\left(5 x \right)}}{5} + \frac{2 \sin{\left(5 x \right)}}{25} + \frac{3 \cos{\left(5 x \right)}}{5}$$
The graph
The answer [src]
  3   cos(5)   2*sin(5)
- - + ------ + --------
  5     5         25   
$$- \frac{3}{5} + \frac{2 \sin{\left(5 \right)}}{25} + \frac{\cos{\left(5 \right)}}{5}$$
=
=
  3   cos(5)   2*sin(5)
- - + ------ + --------
  5     5         25   
$$- \frac{3}{5} + \frac{2 \sin{\left(5 \right)}}{25} + \frac{\cos{\left(5 \right)}}{5}$$
-3/5 + cos(5)/5 + 2*sin(5)/25
Numerical answer [src]
-0.619981504880406
-0.619981504880406

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