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Integral of d{x}:
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Identical expressions
(cosx+18sinx)/(3sinx-4cosx)
( co sinus of e of x plus 18 sinus of x) divide by (3 sinus of x minus 4 co sinus of e of x)
cosx+18sinx/3sinx-4cosx
(cosx+18sinx) divide by (3sinx-4cosx)
(cosx+18sinx)/(3sinx-4cosx)dx
Similar expressions
(cosx+18sinx)/(3sinx+4cosx)
(cosx-18sinx)/(3sinx-4cosx)

Solving integrals/ (cosx+18sinx)/(3sinx-4cosx)

Integral of (cosx+18sinx)/(3sinx-4cosx) dx

The solution

You have entered [src]

 pi                       
 --                       
 2                        
  /                       
 |                        
 |   cos(x) + 18*sin(x)   
 |  ------------------- dx
 |  3*sin(x) - 4*cos(x)   
 |                        
/                         
0

$$\int\limits_{0}^{\frac{\pi}{2}} \frac{18 \sin{\left(x \right)} + \cos{\left(x \right)}}{3 \sin{\left(x \right)} - 4 \cos{\left(x \right)}}\, dx$$

Integral((cos(x) + 18*sin(x))/(3*sin(x) - 4*cos(x)), (x, 0, pi/2))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\frac{18 \sin{\left(x \right)} + \cos{\left(x \right)}}{3 \sin{\left(x \right)} - 4 \cos{\left(x \right)}} = - \frac{18 \sin{\left(x \right)} + \cos{\left(x \right)}}{- 3 \sin{\left(x \right)} + 4 \cos{\left(x \right)}}$
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{18 \sin{\left(x \right)} + \cos{\left(x \right)}}{- 3 \sin{\left(x \right)} + 4 \cos{\left(x \right)}}\right)\, dx = - \int \frac{18 \sin{\left(x \right)} + \cos{\left(x \right)}}{- 3 \sin{\left(x \right)} + 4 \cos{\left(x \right)}}\, dx$
  1. Rewrite the integrand:
    
    $\frac{18 \sin{\left(x \right)} + \cos{\left(x \right)}}{- 3 \sin{\left(x \right)} + 4 \cos{\left(x \right)}} = \frac{18 \sin{\left(x \right)}}{- 3 \sin{\left(x \right)} + 4 \cos{\left(x \right)}} + \frac{\cos{\left(x \right)}}{- 3 \sin{\left(x \right)} + 4 \cos{\left(x \right)}}$
  2. Integrate term-by-term:
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{18 \sin{\left(x \right)}}{- 3 \sin{\left(x \right)} + 4 \cos{\left(x \right)}}\, dx = 18 \int \frac{\sin{\left(x \right)}}{- 3 \sin{\left(x \right)} + 4 \cos{\left(x \right)}}\, dx$
      
      Don't know the steps in finding this integral.
      
      But the integral is
      
      $- \frac{3 x}{25} - \frac{4 \log{\left(- 3 \sin{\left(x \right)} + 4 \cos{\left(x \right)} \right)}}{25}$
      
      So, the result is: $- \frac{54 x}{25} - \frac{72 \log{\left(- 3 \sin{\left(x \right)} + 4 \cos{\left(x \right)} \right)}}{25}$
    1. Don't know the steps in finding this integral.
      
      But the integral is
      
      $\frac{4 x}{25} - \frac{3 \log{\left(- 3 \sin{\left(x \right)} + 4 \cos{\left(x \right)} \right)}}{25}$
    The result is: $- 2 x - 3 \log{\left(- 3 \sin{\left(x \right)} + 4 \cos{\left(x \right)} \right)}$
  So, the result is: $2 x + 3 \log{\left(- 3 \sin{\left(x \right)} + 4 \cos{\left(x \right)} \right)}$
Method #2
1. Rewrite the integrand:
  
  $\frac{18 \sin{\left(x \right)} + \cos{\left(x \right)}}{3 \sin{\left(x \right)} - 4 \cos{\left(x \right)}} = \frac{18 \sin{\left(x \right)}}{3 \sin{\left(x \right)} - 4 \cos{\left(x \right)}} + \frac{\cos{\left(x \right)}}{3 \sin{\left(x \right)} - 4 \cos{\left(x \right)}}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{18 \sin{\left(x \right)}}{3 \sin{\left(x \right)} - 4 \cos{\left(x \right)}}\, dx = 18 \int \frac{\sin{\left(x \right)}}{3 \sin{\left(x \right)} - 4 \cos{\left(x \right)}}\, dx$
    1. Rewrite the integrand:
      
      $\frac{\sin{\left(x \right)}}{3 \sin{\left(x \right)} - 4 \cos{\left(x \right)}} = - \frac{\sin{\left(x \right)}}{- 3 \sin{\left(x \right)} + 4 \cos{\left(x \right)}}$
    2. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{\sin{\left(x \right)}}{- 3 \sin{\left(x \right)} + 4 \cos{\left(x \right)}}\right)\, dx = - \int \frac{\sin{\left(x \right)}}{- 3 \sin{\left(x \right)} + 4 \cos{\left(x \right)}}\, dx$
      
      Don't know the steps in finding this integral.
      
      But the integral is
      
      $- \frac{3 x}{25} - \frac{4 \log{\left(- 3 \sin{\left(x \right)} + 4 \cos{\left(x \right)} \right)}}{25}$
      
      So, the result is: $\frac{3 x}{25} + \frac{4 \log{\left(- 3 \sin{\left(x \right)} + 4 \cos{\left(x \right)} \right)}}{25}$
    So, the result is: $\frac{54 x}{25} + \frac{72 \log{\left(- 3 \sin{\left(x \right)} + 4 \cos{\left(x \right)} \right)}}{25}$
  1. Rewrite the integrand:
    
    $\frac{\cos{\left(x \right)}}{3 \sin{\left(x \right)} - 4 \cos{\left(x \right)}} = - \frac{\cos{\left(x \right)}}{- 3 \sin{\left(x \right)} + 4 \cos{\left(x \right)}}$
  2. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{\cos{\left(x \right)}}{- 3 \sin{\left(x \right)} + 4 \cos{\left(x \right)}}\right)\, dx = - \int \frac{\cos{\left(x \right)}}{- 3 \sin{\left(x \right)} + 4 \cos{\left(x \right)}}\, dx$
    1. Don't know the steps in finding this integral.
      
      But the integral is
      
      $\frac{4 x}{25} - \frac{3 \log{\left(- 3 \sin{\left(x \right)} + 4 \cos{\left(x \right)} \right)}}{25}$
    So, the result is: $- \frac{4 x}{25} + \frac{3 \log{\left(- 3 \sin{\left(x \right)} + 4 \cos{\left(x \right)} \right)}}{25}$
  The result is: $2 x + 3 \log{\left(- 3 \sin{\left(x \right)} + 4 \cos{\left(x \right)} \right)}$
Add the constant of integration:

$2 x + 3 \log{\left(- 3 \sin{\left(x \right)} + 4 \cos{\left(x \right)} \right)}+ \mathrm{constant}$

The answer is:

$2 x + 3 \log{\left(- 3 \sin{\left(x \right)} + 4 \cos{\left(x \right)} \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                              
 |                                                               
 |  cos(x) + 18*sin(x)                                           
 | ------------------- dx = C + 2*x + 3*log(-3*sin(x) + 4*cos(x))
 | 3*sin(x) - 4*cos(x)                                           
 |                                                               
/

$$\int \frac{18 \sin{\left(x \right)} + \cos{\left(x \right)}}{3 \sin{\left(x \right)} - 4 \cos{\left(x \right)}}\, dx = C + 2 x + 3 \log{\left(- 3 \sin{\left(x \right)} + 4 \cos{\left(x \right)} \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

nan

$$\text{NaN}$$

nan

$$\text{NaN}$$

nan

Numerical answer [src]

0.0514600133515972

0.0514600133515972

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

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How to use it?

Identical expressions

Similar expressions

Integral of (cosx+18sinx)/(3sinx-4cosx) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2