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Integral of f(x+4)*sin(x) dx

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The solution

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 |  f*(x + 4)*sin(x) dx
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$$\int\limits_{0}^{1} f \left(x + 4\right) \sin{\left(x \right)}\, dx$$
Integral(f*(x + 4)*sin(x), (x, 0, 1))
Detail solution
  1. The integral of a constant times a function is the constant times the integral of the function:

    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. 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:

        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:

      Method #2

      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:

  2. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
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 | f*(x + 4)*sin(x) dx = C + f*(-4*cos(x) - x*cos(x) + sin(x))
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$$\int f \left(x + 4\right) \sin{\left(x \right)}\, dx = C + f \left(- x \cos{\left(x \right)} + \sin{\left(x \right)} - 4 \cos{\left(x \right)}\right)$$
The answer [src]
4*f + f*(-5*cos(1) + sin(1))
$$f \left(- 5 \cos{\left(1 \right)} + \sin{\left(1 \right)}\right) + 4 f$$
=
=
4*f + f*(-5*cos(1) + sin(1))
$$f \left(- 5 \cos{\left(1 \right)} + \sin{\left(1 \right)}\right) + 4 f$$

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