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Solving integrals/ sin(ln(x))/sqrt(x)

Integral of sin(ln(x))/sqrt(x) dx

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The solution

You have entered [src] 
 oo               
  /               
 |                
 |  sin(log(x))   
 |  ----------- dx
 |       ___      
 |     \/ x       
 |                
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∫0∞sin⁡(log⁡(x))x dx\int\limits_{0}^{\infty} \frac{\sin{\left(\log{\left(x \right)} \right)}}{\sqrt{x}}\, dx 
Integral(sin(log(x))/(sqrt(x)), (x, 0, oo))
Detail solution

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

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

    ∫eu2sin⁡(u) du\int e^{\frac{u}{2}} \sin{\left(u \right)}\, du

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

      1. For the integrand eu2sin⁡(u)e^{\frac{u}{2}} \sin{\left(u \right)}:

        Let u(u)=sin⁡(u)u{\left(u \right)} = \sin{\left(u \right)} and let dv⁡(u)=eu2\operatorname{dv}{\left(u \right)} = e^{\frac{u}{2}}.

        Then ∫eu2sin⁡(u) du=2eu2sin⁡(u)−∫2eu2cos⁡(u) du\int e^{\frac{u}{2}} \sin{\left(u \right)}\, du = 2 e^{\frac{u}{2}} \sin{\left(u \right)} - \int 2 e^{\frac{u}{2}} \cos{\left(u \right)}\, du.

      2. For the integrand 2eu2cos⁡(u)2 e^{\frac{u}{2}} \cos{\left(u \right)}:

        Let u(u)=2cos⁡(u)u{\left(u \right)} = 2 \cos{\left(u \right)} and let dv⁡(u)=eu2\operatorname{dv}{\left(u \right)} = e^{\frac{u}{2}}.

        Then ∫eu2sin⁡(u) du=2eu2sin⁡(u)−4eu2cos⁡(u)+∫(−4eu2sin⁡(u)) du\int e^{\frac{u}{2}} \sin{\left(u \right)}\, du = 2 e^{\frac{u}{2}} \sin{\left(u \right)} - 4 e^{\frac{u}{2}} \cos{\left(u \right)} + \int \left(- 4 e^{\frac{u}{2}} \sin{\left(u \right)}\right)\, du.

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

        5∫eu2sin⁡(u) du=2eu2sin⁡(u)−4eu2cos⁡(u)5 \int e^{\frac{u}{2}} \sin{\left(u \right)}\, du = 2 e^{\frac{u}{2}} \sin{\left(u \right)} - 4 e^{\frac{u}{2}} \cos{\left(u \right)}

        Therefore,

        ∫eu2sin⁡(u) du=2eu2sin⁡(u)5−4eu2cos⁡(u)5\int e^{\frac{u}{2}} \sin{\left(u \right)}\, du = \frac{2 e^{\frac{u}{2}} \sin{\left(u \right)}}{5} - \frac{4 e^{\frac{u}{2}} \cos{\left(u \right)}}{5}

    Now substitute uu back in:

    2xsin⁡(log⁡(x))5−4xcos⁡(log⁡(x))5\frac{2 \sqrt{x} \sin{\left(\log{\left(x \right)} \right)}}{5} - \frac{4 \sqrt{x} \cos{\left(\log{\left(x \right)} \right)}}{5}

  2. Now simplify:

    2x(sin⁡(log⁡(x))−2cos⁡(log⁡(x)))5\frac{2 \sqrt{x} \left(\sin{\left(\log{\left(x \right)} \right)} - 2 \cos{\left(\log{\left(x \right)} \right)}\right)}{5}

  3. Add the constant of integration:

    2x(sin⁡(log⁡(x))−2cos⁡(log⁡(x)))5+constant\frac{2 \sqrt{x} \left(\sin{\left(\log{\left(x \right)} \right)} - 2 \cos{\left(\log{\left(x \right)} \right)}\right)}{5}+ \mathrm{constant}

The answer is:

2x(sin⁡(log⁡(x))−2cos⁡(log⁡(x)))5+constant\frac{2 \sqrt{x} \left(\sin{\left(\log{\left(x \right)} \right)} - 2 \cos{\left(\log{\left(x \right)} \right)}\right)}{5}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                                              
 |                          ___                   ___            
 | sin(log(x))          4*\/ x *cos(log(x))   2*\/ x *sin(log(x))
 | ----------- dx = C - ------------------- + -------------------
 |      ___                      5                     5         
 |    \/ x                                                       
 |                                                               
/
∫sin⁡(log⁡(x))x dx=C+2xsin⁡(log⁡(x))5−4xcos⁡(log⁡(x))5\int \frac{\sin{\left(\log{\left(x \right)} \right)}}{\sqrt{x}}\, dx = C + \frac{2 \sqrt{x} \sin{\left(\log{\left(x \right)} \right)}}{5} - \frac{4 \sqrt{x} \cos{\left(\log{\left(x \right)} \right)}}{5}
Simplify
The answer [src] 
<-oo, oo>
⟨−∞,∞⟩\left\langle -\infty, \infty\right\rangle 
=
= 
<-oo, oo>
⟨−∞,∞⟩\left\langle -\infty, \infty\right\rangle
Упростить

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

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