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Integral of dz÷(5z+tan(y-3x)) dx

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The solution

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  1                      
  /                      
 |                       
 |          1            
 |  ------------------ dz
 |  5*z + tan(y - 3*x)   
 |                       
/                        
0                        
$$\int\limits_{0}^{1} \frac{1}{5 z + \tan{\left(- 3 x + y \right)}}\, dz$$
Integral(1/(5*z + tan(y - 3*x)), (z, 0, 1))
Detail solution
  1. Let .

    Then let and substitute :

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

      1. The integral of is .

      So, the result is:

    Now substitute back in:

  2. Now simplify:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                   
 |                                                    
 |         1                   log(5*z + tan(y - 3*x))
 | ------------------ dz = C + -----------------------
 | 5*z + tan(y - 3*x)                     5           
 |                                                    
/                                                     
$$\int \frac{1}{5 z + \tan{\left(- 3 x + y \right)}}\, dz = C + \frac{\log{\left(5 z + \tan{\left(- 3 x + y \right)} \right)}}{5}$$
The answer [src]
  log(-tan(-y + 3*x))   log(5 - tan(-y + 3*x))
- ------------------- + ----------------------
           5                      5           
$$\frac{\log{\left(5 - \tan{\left(3 x - y \right)} \right)}}{5} - \frac{\log{\left(- \tan{\left(3 x - y \right)} \right)}}{5}$$
=
=
  log(-tan(-y + 3*x))   log(5 - tan(-y + 3*x))
- ------------------- + ----------------------
           5                      5           
$$\frac{\log{\left(5 - \tan{\left(3 x - y \right)} \right)}}{5} - \frac{\log{\left(- \tan{\left(3 x - y \right)} \right)}}{5}$$
-log(-tan(-y + 3*x))/5 + log(5 - tan(-y + 3*x))/5

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

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