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Integral of sin(y)+y*sin(x)+1/x dx

The solution

You have entered [src]

  x                           
  /                           
 |                            
 |  /                    1\   
 |  |sin(y) + y*sin(x) + -| dx
 |  \                    x/   
 |                            
/                             
0

$$\int\limits_{0}^{x} \left(\left(y \sin{\left(x \right)} + \sin{\left(y \right)}\right) + \frac{1}{x}\right)\, dx$$

Integral(sin(y) + y*sin(x) + 1/x, (x, 0, x))

Detail solution

Integrate term-by-term:
1. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int y \sin{\left(x \right)}\, dx = y \int \sin{\left(x \right)}\, dx$
    1. The integral of sine is negative cosine:
      
      $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
    So, the result is: $- y \cos{\left(x \right)}$
  1. The integral of sine is negative cosine:
    
    $\int \sin{\left(y \right)}\, dx = - \cos{\left(y \right)}$
  The result is: $- y \cos{\left(x \right)} - \cos{\left(y \right)}$
1. The integral of $\frac{1}{x}$ is $\log{\left(x \right)}$ .
The result is: $- y \cos{\left(x \right)} + \log{\left(x \right)} - \cos{\left(y \right)}$
Add the constant of integration:

$- y \cos{\left(x \right)} + \log{\left(x \right)} - \cos{\left(y \right)}+ \mathrm{constant}$

The answer is:

$- y \cos{\left(x \right)} + \log{\left(x \right)} - \cos{\left(y \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                           
 |                                                            
 | /                    1\                                    
 | |sin(y) + y*sin(x) + -| dx = C - cos(y) - y*cos(x) + log(x)
 | \                    x/                                    
 |                                                            
/

$$\int \left(\left(y \sin{\left(x \right)} + \sin{\left(y \right)}\right) + \frac{1}{x}\right)\, dx = C - y \cos{\left(x \right)} + \log{\left(x \right)} - \cos{\left(y \right)}$$

Simplify

The answer [src]

oo + x*sin(y) - y*cos(x) + log(x)

$$x \sin{\left(y \right)} - y \cos{\left(x \right)} + \log{\left(x \right)} + \infty$$

oo + x*sin(y) - y*cos(x) + log(x)

$$x \sin{\left(y \right)} - y \cos{\left(x \right)} + \log{\left(x \right)} + \infty$$

oo + x*sin(y) - y*cos(x) + log(x)

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

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Integral of sin(y)+y*sin(x)+1/x dx

Limits of integration:

The graph:

Piecewise:

The solution