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Integral of (x+2)sin5x dx

Limits of integration:

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The graph:

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Piecewise:

The solution

You have entered [src]
  1                    
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 |  (x + 2)*sin(5*x) dx
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$$\int\limits_{0}^{1} \left(x + 2\right) \sin{\left(5 x \right)}\, dx$$
Integral((x + 2)*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. 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:

      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. 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:

      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(5*x)   sin(5*x)   x*cos(5*x)
 | (x + 2)*sin(5*x) dx = C - ---------- + -------- - ----------
 |                               5           25          5     
/                                                              
$$\int \left(x + 2\right) \sin{\left(5 x \right)}\, dx = C - \frac{x \cos{\left(5 x \right)}}{5} + \frac{\sin{\left(5 x \right)}}{25} - \frac{2 \cos{\left(5 x \right)}}{5}$$
The graph
The answer [src]
2   3*cos(5)   sin(5)
- - -------- + ------
5      5         25  
$$- \frac{3 \cos{\left(5 \right)}}{5} + \frac{\sin{\left(5 \right)}}{25} + \frac{2}{5}$$
=
=
2   3*cos(5)   sin(5)
- - -------- + ------
5      5         25  
$$- \frac{3 \cos{\left(5 \right)}}{5} + \frac{\sin{\left(5 \right)}}{25} + \frac{2}{5}$$
2/5 - 3*cos(5)/5 + sin(5)/25
Numerical answer [src]
0.191445717735539
0.191445717735539

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