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Solving integrals/ (x+xy)dy/(yx-y)

Integral of (x+xy)dy/(yx-y) dx

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The solution

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  1           
  /           
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 |  x + x*y   
 |  ------- dy
 |  y*x - y   
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0
∫01xy+xxy−y dy\int\limits_{0}^{1} \frac{x y + x}{x y - y}\, dy 
Integral((x + x*y)/(y*x - y), (y, 0, 1))
Detail solution

  1. There are multiple ways to do this integral.

    Method #1

    1. Rewrite the integrand:

      xy+xxy−y=xx−1+xy(x−1)\frac{x y + x}{x y - y} = \frac{x}{x - 1} + \frac{x}{y \left(x - 1\right)}

    2. Integrate term-by-term:

      1. The integral of a constant is the constant times the variable of integration:

        ∫xx−1 dy=xyx−1\int \frac{x}{x - 1}\, dy = \frac{x y}{x - 1}

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

        ∫xy(x−1) dy=x∫1y dyx−1\int \frac{x}{y \left(x - 1\right)}\, dy = \frac{x \int \frac{1}{y}\, dy}{x - 1}

        1. The integral of 1y\frac{1}{y} is log⁡(y)\log{\left(y \right)}.

        So, the result is: xlog⁡(y)x−1\frac{x \log{\left(y \right)}}{x - 1}

      The result is: xyx−1+xlog⁡(y)x−1\frac{x y}{x - 1} + \frac{x \log{\left(y \right)}}{x - 1}

    Method #2

    1. Rewrite the integrand:

      xy+xxy−y=xyxy−y+xxy−y\frac{x y + x}{x y - y} = \frac{x y}{x y - y} + \frac{x}{x y - y}

    2. Integrate term-by-term:

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

        ∫xyxy−y dy=x∫yxy−y dy\int \frac{x y}{x y - y}\, dy = x \int \frac{y}{x y - y}\, dy

        1. Rewrite the integrand:

          yxy−y=1x−1\frac{y}{x y - y} = \frac{1}{x - 1}

        2. The integral of a constant is the constant times the variable of integration:

          ∫1x−1 dy=yx−1\int \frac{1}{x - 1}\, dy = \frac{y}{x - 1}

        So, the result is: xyx−1\frac{x y}{x - 1}

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

        ∫xxy−y dy=x∫1xy−y dy\int \frac{x}{x y - y}\, dy = x \int \frac{1}{x y - y}\, dy

        1. Rewrite the integrand:

          1xy−y=1y(x−1)\frac{1}{x y - y} = \frac{1}{y \left(x - 1\right)}

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

          ∫1y(x−1) dy=∫1y dyx−1\int \frac{1}{y \left(x - 1\right)}\, dy = \frac{\int \frac{1}{y}\, dy}{x - 1}

          1. The integral of 1y\frac{1}{y} is log⁡(y)\log{\left(y \right)}.

          So, the result is: log⁡(y)x−1\frac{\log{\left(y \right)}}{x - 1}

        So, the result is: xlog⁡(y)x−1\frac{x \log{\left(y \right)}}{x - 1}

      The result is: xyx−1+xlog⁡(y)x−1\frac{x y}{x - 1} + \frac{x \log{\left(y \right)}}{x - 1}

  2. Now simplify:

    x(y+log⁡(y))x−1\frac{x \left(y + \log{\left(y \right)}\right)}{x - 1}

  3. Add the constant of integration:

    x(y+log⁡(y))x−1+constant\frac{x \left(y + \log{\left(y \right)}\right)}{x - 1}+ \mathrm{constant}

The answer is:

x(y+log⁡(y))x−1+constant\frac{x \left(y + \log{\left(y \right)}\right)}{x - 1}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                  
 |                                   
 | x + x*y           x*y     x*log(y)
 | ------- dy = C + ------ + --------
 | y*x - y          -1 + x    -1 + x 
 |                                   
/
∫xy+xxy−y dy=C+xyx−1+xlog⁡(y)x−1\int \frac{x y + x}{x y - y}\, dy = C + \frac{x y}{x - 1} + \frac{x \log{\left(y \right)}}{x - 1}
Simplify
The answer [src] 
       /  x   \     x   
oo*sign|------| + ------
       \-1 + x/   -1 + x
xx−1+∞sign⁡(xx−1)\frac{x}{x - 1} + \infty \operatorname{sign}{\left(\frac{x}{x - 1} \right)} 
=
= 
       /  x   \     x   
oo*sign|------| + ------
       \-1 + x/   -1 + x
xx−1+∞sign⁡(xx−1)\frac{x}{x - 1} + \infty \operatorname{sign}{\left(\frac{x}{x - 1} \right)} 
oo*sign(x/(-1 + x)) + x/(-1 + x)

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

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