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

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The solution

You have entered [src]
 oo               
  /               
 |                
 |  sin(log(x))   
 |  ----------- dx
 |       ___      
 |     \/ x       
 |                
/                 
0                 
0sin(log(x))xdx\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:

        5eu2sin(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)54eu2cos(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))54xcos(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))xdx=C+2xsin(log(x))54xcos(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}
The answer [src]
<-oo, oo>
,\left\langle -\infty, \infty\right\rangle
=
=
<-oo, oo>
,\left\langle -\infty, \infty\right\rangle

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