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sin(x)/(5+3sin(x))

Integral of sin(x)/(5+3sin(x)) dx

Limits of integration:

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The graph:

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The solution

You have entered [src]
  1                
  /                
 |                 
 |     sin(x)      
 |  ------------ dx
 |  5 + 3*sin(x)   
 |                 
/                  
0                  
$$\int\limits_{0}^{1} \frac{\sin{\left(x \right)}}{3 \sin{\left(x \right)} + 5}\, dx$$
Integral(sin(x)/(5 + 3*sin(x)), (x, 0, 1))
The answer (Indefinite) [src]
                               /         /x\\                 /x   pi\
                               |    5*tan|-||                 |- - --|
  /                            |3        \2/|                 |2   2 |
 |                       5*atan|- + --------|       5*pi*floor|------|
 |    sin(x)                   \4      4    /   x             \  pi  /
 | ------------ dx = C - -------------------- + - - ------------------
 | 5 + 3*sin(x)                   6             3           6         
 |                                                                    
/                                                                     
$$\int \frac{\sin{\left(x \right)}}{3 \sin{\left(x \right)} + 5}\, dx = C + \frac{x}{3} - \frac{5 \operatorname{atan}{\left(\frac{5 \tan{\left(\frac{x}{2} \right)}}{4} + \frac{3}{4} \right)}}{6} - \frac{5 \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor}{6}$$
The graph
The answer [src]
          /3   5*tan(1/2)\              
    5*atan|- + ----------|              
1         \4       4     /   5*atan(3/4)
- - ---------------------- + -----------
3             6                   6     
$$- \frac{5 \operatorname{atan}{\left(\frac{5 \tan{\left(\frac{1}{2} \right)}}{4} + \frac{3}{4} \right)}}{6} + \frac{1}{3} + \frac{5 \operatorname{atan}{\left(\frac{3}{4} \right)}}{6}$$
=
=
          /3   5*tan(1/2)\              
    5*atan|- + ----------|              
1         \4       4     /   5*atan(3/4)
- - ---------------------- + -----------
3             6                   6     
$$- \frac{5 \operatorname{atan}{\left(\frac{5 \tan{\left(\frac{1}{2} \right)}}{4} + \frac{3}{4} \right)}}{6} + \frac{1}{3} + \frac{5 \operatorname{atan}{\left(\frac{3}{4} \right)}}{6}$$
1/3 - 5*atan(3/4 + 5*tan(1/2)/4)/6 + 5*atan(3/4)/6
Numerical answer [src]
0.0683477626031842
0.0683477626031842
The graph
Integral of sin(x)/(5+3sin(x)) dx

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