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sinxcos(cosx-1)
Solving integrals/ sinxcos(cosx-1)

Integral of sinxcos(cosx-1) dx

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∫01sin⁡(x)cos⁡(cos⁡(x)−1) dx\int\limits_{0}^{1} \sin{\left(x \right)} \cos{\left(\cos{\left(x \right)} - 1 \right)}\, dx 
Integral(sin(x)*cos(cos(x) - 1*1), (x, 0, 1))
Detail solution

  1. There are multiple ways to do this integral.

    Method #1

    1. Let u=cos⁡(x)−1u = \cos{\left(x \right)} - 1.

      Then let du=−sin⁡(x)dxdu = - \sin{\left(x \right)} dx and substitute −du- du:

      ∫cos⁡(u) du\int \cos{\left(u \right)}\, du

      1. The integral of a constant times a function is the constant times the integral of the function:

        ∫(−cos⁡(u)) du=−∫cos⁡(u) du\int \left(- \cos{\left(u \right)}\right)\, du = - \int \cos{\left(u \right)}\, du

        1. The integral of cosine is sine:

          ∫cos⁡(u) du=sin⁡(u)\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}

        So, the result is: −sin⁡(u)- \sin{\left(u \right)}

      Now substitute uu back in:

      −sin⁡(cos⁡(x)−1)- \sin{\left(\cos{\left(x \right)} - 1 \right)}

    Method #2

    1. Rewrite the integrand:

      sin⁡(x)cos⁡(cos⁡(x)−1)=sin⁡(1)sin⁡(x)sin⁡(cos⁡(x))+sin⁡(x)cos⁡(1)cos⁡(cos⁡(x))\sin{\left(x \right)} \cos{\left(\cos{\left(x \right)} - 1 \right)} = \sin{\left(1 \right)} \sin{\left(x \right)} \sin{\left(\cos{\left(x \right)} \right)} + \sin{\left(x \right)} \cos{\left(1 \right)} \cos{\left(\cos{\left(x \right)} \right)}

    2. Integrate term-by-term:

      1. The integral of a constant times a function is the constant times the integral of the function:

        ∫sin⁡(1)sin⁡(x)sin⁡(cos⁡(x)) dx=sin⁡(1)∫sin⁡(x)sin⁡(cos⁡(x)) dx\int \sin{\left(1 \right)} \sin{\left(x \right)} \sin{\left(\cos{\left(x \right)} \right)}\, dx = \sin{\left(1 \right)} \int \sin{\left(x \right)} \sin{\left(\cos{\left(x \right)} \right)}\, dx

        1. Let u=cos⁡(x)u = \cos{\left(x \right)}.

          Then let du=−sin⁡(x)dxdu = - \sin{\left(x \right)} dx and substitute −du- du:

          ∫sin⁡(u) du\int \sin{\left(u \right)}\, du

          1. The integral of a constant times a function is the constant times the integral of the function:

            ∫(−sin⁡(u)) du=−∫sin⁡(u) du\int \left(- \sin{\left(u \right)}\right)\, du = - \int \sin{\left(u \right)}\, du

            1. The integral of sine is negative cosine:

              ∫sin⁡(u) du=−cos⁡(u)\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}

            So, the result is: cos⁡(u)\cos{\left(u \right)}

          Now substitute uu back in:

          cos⁡(cos⁡(x))\cos{\left(\cos{\left(x \right)} \right)}

        So, the result is: sin⁡(1)cos⁡(cos⁡(x))\sin{\left(1 \right)} \cos{\left(\cos{\left(x \right)} \right)}

      1. The integral of a constant times a function is the constant times the integral of the function:

        ∫sin⁡(x)cos⁡(1)cos⁡(cos⁡(x)) dx=cos⁡(1)∫sin⁡(x)cos⁡(cos⁡(x)) dx\int \sin{\left(x \right)} \cos{\left(1 \right)} \cos{\left(\cos{\left(x \right)} \right)}\, dx = \cos{\left(1 \right)} \int \sin{\left(x \right)} \cos{\left(\cos{\left(x \right)} \right)}\, dx

        1. Let u=cos⁡(x)u = \cos{\left(x \right)}.

          Then let du=−sin⁡(x)dxdu = - \sin{\left(x \right)} dx and substitute −du- du:

          ∫cos⁡(u) du\int \cos{\left(u \right)}\, du

          1. The integral of a constant times a function is the constant times the integral of the function:

            ∫(−cos⁡(u)) du=−∫cos⁡(u) du\int \left(- \cos{\left(u \right)}\right)\, du = - \int \cos{\left(u \right)}\, du

            1. The integral of cosine is sine:

              ∫cos⁡(u) du=sin⁡(u)\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}

            So, the result is: −sin⁡(u)- \sin{\left(u \right)}

          Now substitute uu back in:

          −sin⁡(cos⁡(x))- \sin{\left(\cos{\left(x \right)} \right)}

        So, the result is: −sin⁡(cos⁡(x))cos⁡(1)- \sin{\left(\cos{\left(x \right)} \right)} \cos{\left(1 \right)}

      The result is: −sin⁡(cos⁡(x))cos⁡(1)+sin⁡(1)cos⁡(cos⁡(x))- \sin{\left(\cos{\left(x \right)} \right)} \cos{\left(1 \right)} + \sin{\left(1 \right)} \cos{\left(\cos{\left(x \right)} \right)}

  2. Now simplify:

    −sin⁡(cos⁡(x)−1)- \sin{\left(\cos{\left(x \right)} - 1 \right)}

  3. Add the constant of integration:

    −sin⁡(cos⁡(x)−1)+constant- \sin{\left(\cos{\left(x \right)} - 1 \right)}+ \mathrm{constant}

The answer is:

−sin⁡(cos⁡(x)−1)+constant- \sin{\left(\cos{\left(x \right)} - 1 \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                               
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 | sin(x)*cos(cos(x) - 1) dx = C - sin(cos(x) - 1)
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−sin⁡(cos⁡x−1)-\sin \left(\cos x-1\right)
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.900.01.0
Plot the graph f(x)
The answer [src] 
sin(1 - cos(1))
−sin⁡(cos⁡1−1)-\sin \left(\cos 1-1\right) 
=
= 
sin(1 - cos(1))
sin⁡(−cos⁡(1)+1)\sin{\left(- \cos{\left(1 \right)} + 1 \right)}
Simplify
Numerical answer [src] 
0.443677204755532
0.443677204755532
The graph
Integral of sinxcos(cosx-1) dx

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

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