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Integral of 1/(2sin(x)-cos(x)+3) dx

Limits of integration:

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The graph:

from to

Piecewise:

The solution

You have entered [src]
  1                         
  /                         
 |                          
 |            1             
 |  --------------------- dx
 |  2*sin(x) - cos(x) + 3   
 |                          
/                           
0                           
$$\int\limits_{0}^{1} \frac{1}{\left(2 \sin{\left(x \right)} - \cos{\left(x \right)}\right) + 3}\, dx$$
Integral(1/(2*sin(x) - cos(x) + 3), (x, 0, 1))
Detail solution
  1. Rewrite the integrand:

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

    1. Don't know the steps in finding this integral.

      But the integral is

    So, the result is:

  3. Now simplify:

  4. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                       /x   pi\                     
 |                                        |- - --|                     
 |           1                            |2   2 |       /         /x\\
 | --------------------- dx = C + pi*floor|------| + atan|1 + 2*tan|-||
 | 2*sin(x) - cos(x) + 3                  \  pi  /       \         \2//
 |                                                                     
/                                                                      
$$\int \frac{1}{\left(2 \sin{\left(x \right)} - \cos{\left(x \right)}\right) + 3}\, dx = C + \operatorname{atan}{\left(2 \tan{\left(\frac{x}{2} \right)} + 1 \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor$$
The graph
The answer [src]
  pi                       
- -- + atan(1 + 2*tan(1/2))
  4                        
$$- \frac{\pi}{4} + \operatorname{atan}{\left(1 + 2 \tan{\left(\frac{1}{2} \right)} \right)}$$
=
=
  pi                       
- -- + atan(1 + 2*tan(1/2))
  4                        
$$- \frac{\pi}{4} + \operatorname{atan}{\left(1 + 2 \tan{\left(\frac{1}{2} \right)} \right)}$$
-pi/4 + atan(1 + 2*tan(1/2))
Numerical answer [src]
0.33960810241123
0.33960810241123

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