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1/(2+sinx)
Solving integrals/ 1/(2+sinx)

Integral of 1/(2+sinx) dx

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The solution

You have entered [src] 
  1              
  /              
 |               
 |      1        
 |  ---------- dx
 |  2 + sin(x)   
 |               
/                
0
$$\int\limits_{0}^{1} \frac{1}{\sin{\left(x \right)} + 2}\, dx$$ 
Integral(1/(2 + sin(x)), (x, 0, 1))
The answer (Indefinite) [src] 
                               /        /x   pi\       /            ___    /x\\\
                               |        |- - --|       |  ___   2*\/ 3 *tan|-|||
  /                        ___ |        |2   2 |       |\/ 3               \2/||
 |                     2*\/ 3 *|pi*floor|------| + atan|----- + --------------||
 |     1                       \        \  pi  /       \  3           3       //
 | ---------- dx = C + ---------------------------------------------------------
 | 2 + sin(x)                                      3                            
 |                                                                              
/
$$\int \frac{1}{\sin{\left(x \right)} + 2}\, dx = C + \frac{2 \sqrt{3} \left(\operatorname{atan}{\left(\frac{2 \sqrt{3} \tan{\left(\frac{x}{2} \right)}}{3} + \frac{\sqrt{3}}{3} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{3}$$
Simplify
The graph
Plot the graph f(x)
The answer [src] 
        /          /  ___       ___         \\             
    ___ |          |\/ 3    2*\/ 3 *tan(1/2)||             
2*\/ 3 *|-pi + atan|----- + ----------------||          ___
        \          \  3            3        //   5*pi*\/ 3 
---------------------------------------------- + ----------
                      3                              9
$$\frac{2 \sqrt{3} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{3}}{3} + \frac{2 \sqrt{3} \tan{\left(\frac{1}{2} \right)}}{3} \right)}\right)}{3} + \frac{5 \sqrt{3} \pi}{9}$$ 
=
= 
        /          /  ___       ___         \\             
    ___ |          |\/ 3    2*\/ 3 *tan(1/2)||             
2*\/ 3 *|-pi + atan|----- + ----------------||          ___
        \          \  3            3        //   5*pi*\/ 3 
---------------------------------------------- + ----------
                      3                              9
$$\frac{2 \sqrt{3} \left(- \pi + \operatorname{atan}{\left(\frac{\sqrt{3}}{3} + \frac{2 \sqrt{3} \tan{\left(\frac{1}{2} \right)}}{3} \right)}\right)}{3} + \frac{5 \sqrt{3} \pi}{9}$$ 
2*sqrt(3)*(-pi + atan(sqrt(3)/3 + 2*sqrt(3)*tan(1/2)/3))/3 + 5*pi*sqrt(3)/9
Numerical answer [src] 
0.410833926083987
0.410833926083987
The graph
Integral of 1/(2+sinx) dx

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

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