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

The solution

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  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

Rewrite the integrand:

$\frac{1}{\left(2 \sin{\left(x \right)} - \cos{\left(x \right)}\right) + 3} = - \frac{1}{- 2 \sin{\left(x \right)} + \cos{\left(x \right)} - 3}$
The integral of a constant times a function is the constant times the integral of the function:

$\int \left(- \frac{1}{- 2 \sin{\left(x \right)} + \cos{\left(x \right)} - 3}\right)\, dx = - \int \frac{1}{- 2 \sin{\left(x \right)} + \cos{\left(x \right)} - 3}\, dx$
1. Don't know the steps in finding this integral.
  
  But the integral is
  
  $- \operatorname{atan}{\left(2 \tan{\left(\frac{x}{2} \right)} + 1 \right)} - \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor$
So, the result is: $\operatorname{atan}{\left(2 \tan{\left(\frac{x}{2} \right)} + 1 \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor$
Now simplify:

$\operatorname{atan}{\left(2 \tan{\left(\frac{x}{2} \right)} + 1 \right)} + \pi \left\lfloor{\frac{x}{2 \pi} - \frac{1}{2}}\right\rfloor$
Add the constant of integration:

$\operatorname{atan}{\left(2 \tan{\left(\frac{x}{2} \right)} + 1 \right)} + \pi \left\lfloor{\frac{x}{2 \pi} - \frac{1}{2}}\right\rfloor+ \mathrm{constant}$

The answer is:

$\operatorname{atan}{\left(2 \tan{\left(\frac{x}{2} \right)} + 1 \right)} + \pi \left\lfloor{\frac{x}{2 \pi} - \frac{1}{2}}\right\rfloor+ \mathrm{constant}$

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//
 |                                                                     
/

$$\int \frac{1}{\left(2 \sin{\left(x \right)} - \cos{\left(x \right)}\right) + 3}\, dx = C + \operatorname{atan}{\left(2 \tan{\left(\frac{x}{2} \right)} + 1 \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor$$

Simplify

The graph

Plot the graph f(x)

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.

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

Limits of integration:

The graph:

Piecewise:

The solution