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Solving integrals/ cos(log(x))

Integral of cos(log(x)) dx

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The solution

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

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

    Then let du=dxxdu = \frac{dx}{x} and substitute dudu:

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

    1. Use integration by parts, noting that the integrand eventually repeats itself.

      1. For the integrand eucos⁡(u)e^{u} \cos{\left(u \right)}:

        Let u(u)=cos⁡(u)u{\left(u \right)} = \cos{\left(u \right)} and let dv⁡(u)=eu\operatorname{dv}{\left(u \right)} = e^{u}.

        Then ∫eucos⁡(u) du=eucos⁡(u)−∫(−eusin⁡(u)) du\int e^{u} \cos{\left(u \right)}\, du = e^{u} \cos{\left(u \right)} - \int \left(- e^{u} \sin{\left(u \right)}\right)\, du.

      2. For the integrand −eusin⁡(u)- e^{u} \sin{\left(u \right)}:

        Let u(u)=−sin⁡(u)u{\left(u \right)} = - \sin{\left(u \right)} and let dv⁡(u)=eu\operatorname{dv}{\left(u \right)} = e^{u}.

        Then ∫eucos⁡(u) du=eusin⁡(u)+eucos⁡(u)+∫(−eucos⁡(u)) du\int e^{u} \cos{\left(u \right)}\, du = e^{u} \sin{\left(u \right)} + e^{u} \cos{\left(u \right)} + \int \left(- e^{u} \cos{\left(u \right)}\right)\, du.

      3. Notice that the integrand has repeated itself, so move it to one side:

        2∫eucos⁡(u) du=eusin⁡(u)+eucos⁡(u)2 \int e^{u} \cos{\left(u \right)}\, du = e^{u} \sin{\left(u \right)} + e^{u} \cos{\left(u \right)}

        Therefore,

        ∫eucos⁡(u) du=eusin⁡(u)2+eucos⁡(u)2\int e^{u} \cos{\left(u \right)}\, du = \frac{e^{u} \sin{\left(u \right)}}{2} + \frac{e^{u} \cos{\left(u \right)}}{2}

    Now substitute uu back in:

    xsin⁡(log⁡(x))2+xcos⁡(log⁡(x))2\frac{x \sin{\left(\log{\left(x \right)} \right)}}{2} + \frac{x \cos{\left(\log{\left(x \right)} \right)}}{2}

  2. Now simplify:

    2xsin⁡(log⁡(x)+π4)2\frac{\sqrt{2} x \sin{\left(\log{\left(x \right)} + \frac{\pi}{4} \right)}}{2}

  3. Add the constant of integration:

    2xsin⁡(log⁡(x)+π4)2+constant\frac{\sqrt{2} x \sin{\left(\log{\left(x \right)} + \frac{\pi}{4} \right)}}{2}+ \mathrm{constant}

The answer is:

2xsin⁡(log⁡(x)+π4)2+constant\frac{\sqrt{2} x \sin{\left(\log{\left(x \right)} + \frac{\pi}{4} \right)}}{2}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                                  
 |                      x*cos(log(x))   x*sin(log(x))
 | cos(log(x)) dx = C + ------------- + -------------
 |                            2               2      
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x (sin⁡log⁡x+cos⁡log⁡x)2{{x\,\left(\sin \log x+\cos \log x\right)}\over{2}}
Simplify
The answer [src] 
1/2
12{{1}\over{2}} 
=
= 
1/2
12\frac{1}{2}
Упростить
Numerical answer [src] 
0.5
0.5

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

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