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Solving integrals/ z-1/(x*y*z^2)

Integral of z-1/(x*y*z^2) dz

Limits of integration:

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The solution

You have entered [src] 
  z                
  /                
 |                 
 |  /      1   \   
 |  |z - ------| dz
 |  |         2|   
 |  \    x*y*z /   
 |                 
/                  
0
∫0z(z−1z2xy) dz\int\limits_{0}^{z} \left(z - \frac{1}{z^{2} x y}\right)\, dz 
Integral(z - 1/((x*y)*z^2), (z, 0, z))
Detail solution

  1. Integrate term-by-term:

    1. The integral of znz^{n} is zn+1n+1\frac{z^{n + 1}}{n + 1} when n≠−1n \neq -1:

      ∫z dz=z22\int z\, dz = \frac{z^{2}}{2}

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

      ∫(−1z2xy) dz=−∫1z2xy dz\int \left(- \frac{1}{z^{2} x y}\right)\, dz = - \int \frac{1}{z^{2} x y}\, dz

      1. The integral of 1z2+1\frac{1}{z^{2} + 1} is ∞~atan⁡(∞~z)xy\frac{\tilde{\infty} \operatorname{atan}{\left(\tilde{\infty} z \right)}}{x y}.

      So, the result is: ∞~atan⁡(∞~z)xy\frac{\tilde{\infty} \operatorname{atan}{\left(\tilde{\infty} z \right)}}{x y}

    The result is: z22+∞~atan⁡(∞~z)xy\frac{z^{2}}{2} + \frac{\tilde{\infty} \operatorname{atan}{\left(\tilde{\infty} z \right)}}{x y}

  2. Add the constant of integration:

    z22+∞~atan⁡(∞~z)xy+constant\frac{z^{2}}{2} + \frac{\tilde{\infty} \operatorname{atan}{\left(\tilde{\infty} z \right)}}{x y}+ \mathrm{constant}

The answer is:

z22+∞~atan⁡(∞~z)xy+constant\frac{z^{2}}{2} + \frac{\tilde{\infty} \operatorname{atan}{\left(\tilde{\infty} z \right)}}{x y}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                          
 |                        2                  
 | /      1   \          z    zoo*atan(zoo*z)
 | |z - ------| dz = C + -- + ---------------
 | |         2|          2          x*y      
 | \    x*y*z /                              
 |                                           
/
∫(z−1z2xy) dz=C+z22+∞~atan⁡(∞~z)xy\int \left(z - \frac{1}{z^{2} x y}\right)\, dz = C + \frac{z^{2}}{2} + \frac{\tilde{\infty} \operatorname{atan}{\left(\tilde{\infty} z \right)}}{x y}
Simplify
The answer [src] 
                          2
                 1   x*y*z 
                 - + ------
         / 1 \   z     2   
- oo*sign|---| + ----------
         \x*y/      x*y
−∞sign⁡(1xy)+xyz22+1zxy- \infty \operatorname{sign}{\left(\frac{1}{x y} \right)} + \frac{\frac{x y z^{2}}{2} + \frac{1}{z}}{x y} 
=
= 
                          2
                 1   x*y*z 
                 - + ------
         / 1 \   z     2   
- oo*sign|---| + ----------
         \x*y/      x*y
−∞sign⁡(1xy)+xyz22+1zxy- \infty \operatorname{sign}{\left(\frac{1}{x y} \right)} + \frac{\frac{x y z^{2}}{2} + \frac{1}{z}}{x y} 
-oo*sign(1/(x*y)) + (1/z + x*y*z^2/2)/(x*y)

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

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