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

The solution

You have entered [src]

 oo               
  /               
 |                
 |  sin(log(x))   
 |  ----------- dx
 |       ___      
 |     \/ x       
 |                
/                 
0

$$\int\limits_{0}^{\infty} \frac{\sin{\left(\log{\left(x \right)} \right)}}{\sqrt{x}}\, dx$$

Integral(sin(log(x))/(sqrt(x)), (x, 0, oo))

Detail solution

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

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

$\int e^{\frac{u}{2}} \sin{\left(u \right)}\, du$
1. Use integration by parts, noting that the integrand eventually repeats itself.
  1. For the integrand $e^{\frac{u}{2}} \sin{\left(u \right)}$ :
    
    Let $u{\left(u \right)} = \sin{\left(u \right)}$ and let $\operatorname{dv}{\left(u \right)} = e^{\frac{u}{2}}$ .
    
    Then $\int e^{\frac{u}{2}} \sin{\left(u \right)}\, du = 2 e^{\frac{u}{2}} \sin{\left(u \right)} - \int 2 e^{\frac{u}{2}} \cos{\left(u \right)}\, du$ .
  2. For the integrand $2 e^{\frac{u}{2}} \cos{\left(u \right)}$ :
    
    Let $u{\left(u \right)} = 2 \cos{\left(u \right)}$ and let $\operatorname{dv}{\left(u \right)} = e^{\frac{u}{2}}$ .
    
    Then $\int e^{\frac{u}{2}} \sin{\left(u \right)}\, du = 2 e^{\frac{u}{2}} \sin{\left(u \right)} - 4 e^{\frac{u}{2}} \cos{\left(u \right)} + \int \left(- 4 e^{\frac{u}{2}} \sin{\left(u \right)}\right)\, du$ .
  3. Notice that the integrand has repeated itself, so move it to one side:
    
    $5 \int e^{\frac{u}{2}} \sin{\left(u \right)}\, du = 2 e^{\frac{u}{2}} \sin{\left(u \right)} - 4 e^{\frac{u}{2}} \cos{\left(u \right)}$
    
    Therefore,
    
    $\int e^{\frac{u}{2}} \sin{\left(u \right)}\, du = \frac{2 e^{\frac{u}{2}} \sin{\left(u \right)}}{5} - \frac{4 e^{\frac{u}{2}} \cos{\left(u \right)}}{5}$
Now substitute $u$ back in:

$\frac{2 \sqrt{x} \sin{\left(\log{\left(x \right)} \right)}}{5} - \frac{4 \sqrt{x} \cos{\left(\log{\left(x \right)} \right)}}{5}$
Now simplify:

$\frac{2 \sqrt{x} \left(\sin{\left(\log{\left(x \right)} \right)} - 2 \cos{\left(\log{\left(x \right)} \right)}\right)}{5}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                                                              
 |                          ___                   ___            
 | sin(log(x))          4*\/ x *cos(log(x))   2*\/ x *sin(log(x))
 | ----------- dx = C - ------------------- + -------------------
 |      ___                      5                     5         
 |    \/ x                                                       
 |                                                               
/

$$\int \frac{\sin{\left(\log{\left(x \right)} \right)}}{\sqrt{x}}\, dx = C + \frac{2 \sqrt{x} \sin{\left(\log{\left(x \right)} \right)}}{5} - \frac{4 \sqrt{x} \cos{\left(\log{\left(x \right)} \right)}}{5}$$

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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Identical expressions

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

Limits of integration:

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The solution