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dz/z(z^ two + four)
dz divide by z(z squared plus 4)
dz divide by z(z to the power of two plus four)
dz/z(z2+4)
dz/zz2+4
dz/z(z²+4)
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Similar expressions
dz/z(z^2-4)

Integral of dz/z(z^2+4) dx

The solution

You have entered [src]

    1                 
    /                 
   |                  
   |     1 / 2    \   
   |   1*-*\z  + 4/ dz
   |     z            
   |                  
  /                   
-1 + z

$$\int\limits_{z - 1}^{1} 1 \cdot \frac{1}{z} \left(z^{2} + 4\right)\, dz$$

Integral(1*(z^2 + 4)/z, (z, -1 + z, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = z^{2}$ .
  
  Then let $du = 2 z dz$ and substitute $\frac{du}{2}$ :
  
  $\int \frac{u + 4}{4 u}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{u + 4}{2 u}\, du = \frac{\int \frac{u + 4}{u}\, du}{2}$
    1. Rewrite the integrand:
      
      $\frac{u + 4}{u} = 1 + \frac{4}{u}$
    2. Integrate term-by-term:
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{4}{u}\, du = 4 \int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      So, the result is: $4 \log{\left(u \right)}$
      
      The result is: $u + 4 \log{\left(u \right)}$
    So, the result is: $\frac{u}{2} + 2 \log{\left(u \right)}$
  Now substitute $u$ back in:
  
  $\frac{z^{2}}{2} + 2 \log{\left(z^{2} \right)}$
Method #2
1. Rewrite the integrand:
  
  $1 \cdot \frac{1}{z} \left(z^{2} + 4\right) = z + \frac{4}{z}$
2. Integrate term-by-term:
  1. The integral of $z^{n}$ is $\frac{z^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int z\, dz = \frac{z^{2}}{2}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{4}{z}\, dz = 4 \int \frac{1}{z}\, dz$
    1. The integral of $\frac{1}{z}$ is $\log{\left(z \right)}$ .
    So, the result is: $4 \log{\left(z \right)}$
  The result is: $\frac{z^{2}}{2} + 4 \log{\left(z \right)}$
Add the constant of integration:

$\frac{z^{2}}{2} + 2 \log{\left(z^{2} \right)}+ \mathrm{constant}$

The answer is:

$\frac{z^{2}}{2} + 2 \log{\left(z^{2} \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                    
 |                        2            
 |   1 / 2    \          z         / 2\
 | 1*-*\z  + 4/ dz = C + -- + 2*log\z /
 |   z                   2             
 |                                     
/

$$\int 1 \cdot \frac{1}{z} \left(z^{2} + 4\right)\, dz = C + \frac{z^{2}}{2} + 2 \log{\left(z^{2} \right)}$$

Simplify

The answer [src]

                            2
1                   (-1 + z) 
- - 4*log(-1 + z) - ---------
2                       2

$$- \frac{\left(z - 1\right)^{2}}{2} - 4 \log{\left(z - 1 \right)} + \frac{1}{2}$$

                            2
1                   (-1 + z) 
- - 4*log(-1 + z) - ---------
2                       2

$$- \frac{\left(z - 1\right)^{2}}{2} - 4 \log{\left(z - 1 \right)} + \frac{1}{2}$$

Упростить

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

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Identical expressions

Similar expressions

Integral of dz/z(z^2+4) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2