Mister Exam

Integral of dz÷(5z+tan(y-3x)) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1                        
  /                        
 |                         
 |            1            
 |  1*------------------ dz
 |    5*z + tan(y - 3*x)   
 |                         
/                          
0                          
$$\int\limits_{0}^{1} 1 \cdot \frac{1}{5 z + \tan{\left(- 3 x + y \right)}}\, dz$$
Integral(1/(5*z + tan(y - 3*x)), (z, 0, 1))
Detail solution
  1. There are multiple ways to do this integral.

    Method #1

    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:

    Method #2

    1. Rewrite the integrand:

    2. 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:

    Method #3

    1. Rewrite the integrand:

    2. 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))
 | 1*------------------ dz = C + -----------------------
 |   5*z + tan(y - 3*x)                     5           
 |                                                      
/                                                       
$${{\log \left(5\,z+\tan \left(y-3\,x\right)\right)}\over{5}}$$
The answer [src]
  log(-tan(-y + 3*x))   log(5 - tan(-y + 3*x))
- ------------------- + ----------------------
           5                      5           
$$\frac{\log{\left(- \tan{\left(3 x - y \right)} + 5 \right)}}{5} - \frac{\log{\left(- \tan{\left(3 x - y \right)} \right)}}{5}$$
=
=
  log(-tan(-y + 3*x))   log(5 - tan(-y + 3*x))
- ------------------- + ----------------------
           5                      5           
$$\frac{\log{\left(- \tan{\left(3 x - y \right)} + 5 \right)}}{5} - \frac{\log{\left(- \tan{\left(3 x - y \right)} \right)}}{5}$$

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