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Solving integrals/ cos(ln(x))/x

Integral of cos(ln(x))/x dx

Limits of integration:

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The solution

You have entered [src] 
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∫01cos⁡(log⁡(x))x dx\int\limits_{0}^{1} \frac{\cos{\left(\log{\left(x \right)} \right)}}{x}\, dx 
Integral(cos(log(x))/x, (x, 0, 1))
Detail solution

  1. There are multiple ways to do this integral.

    Method #1

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

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

      ∫cos⁡(u) 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)}

      Now substitute uu back in:

      sin⁡(log⁡(x))\sin{\left(\log{\left(x \right)} \right)}

    Method #2

    1. Let u=1xu = \frac{1}{x}.

      Then let du=−dxx2du = - \frac{dx}{x^{2}} and substitute −du- du:

      ∫(−cos⁡(log⁡(1u))u) du\int \left(- \frac{\cos{\left(\log{\left(\frac{1}{u} \right)} \right)}}{u}\right)\, du

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

        ∫cos⁡(log⁡(1u))u du=−∫cos⁡(log⁡(1u))u du\int \frac{\cos{\left(\log{\left(\frac{1}{u} \right)} \right)}}{u}\, du = - \int \frac{\cos{\left(\log{\left(\frac{1}{u} \right)} \right)}}{u}\, du

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

          Then let du=−duudu = - \frac{du}{u} and substitute −du- du:

          ∫(−cos⁡(u)) du\int \left(- \cos{\left(u \right)}\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 \cos{\left(u \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⁡(log⁡(1u))- \sin{\left(\log{\left(\frac{1}{u} \right)} \right)}

        So, the result is: sin⁡(log⁡(1u))\sin{\left(\log{\left(\frac{1}{u} \right)} \right)}

      Now substitute uu back in:

      sin⁡(log⁡(x))\sin{\left(\log{\left(x \right)} \right)}

  2. Add the constant of integration:

    sin⁡(log⁡(x))+constant\sin{\left(\log{\left(x \right)} \right)}+ \mathrm{constant}

The answer is:

sin⁡(log⁡(x))+constant\sin{\left(\log{\left(x \right)} \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                
 |                                 
 | cos(log(x))                     
 | ----------- dx = C + sin(log(x))
 |      x                          
 |                                 
/
∫cos⁡(log⁡(x))x dx=C+sin⁡(log⁡(x))\int \frac{\cos{\left(\log{\left(x \right)} \right)}}{x}\, dx = C + \sin{\left(\log{\left(x \right)} \right)}
Simplify
The answer [src] 
<-1, 1>
⟨−1,1⟩\left\langle -1, 1\right\rangle 
=
= 
<-1, 1>
⟨−1,1⟩\left\langle -1, 1\right\rangle 
AccumBounds(-1, 1)
Numerical answer [src] 
0.110056905018961
0.110056905018961

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

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