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Identical expressions
(sinx+cosx)/(three +sin2x)
( sinus of x plus co sinus of e of x) divide by (3 plus sinus of 2x)
( sinus of x plus co sinus of e of x) divide by (three plus sinus of 2x)
sinx+cosx/3+sin2x
(sinx+cosx) divide by (3+sin2x)
(sinx+cosx)/(3+sin2x)dx
Similar expressions
(sinx-cosx)/(3+sin2x)
(sinx+cosx)/(3-sin2x)

Solving integrals/ (sinx+cosx)/(3+sin2x)

Integral of (sinx+cosx)/(3+sin2x) dx

The solution

You have entered [src]

  1                   
  /                   
 |                    
 |  sin(x) + cos(x)   
 |  --------------- dx
 |    3 + sin(2*x)    
 |                    
/                     
0

$$\int\limits_{0}^{1} \frac{\sin{\left(x \right)} + \cos{\left(x \right)}}{\sin{\left(2 x \right)} + 3}\, dx$$

Integral((sin(x) + cos(x))/(3 + sin(2*x)), (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\frac{\sin{\left(x \right)} + \cos{\left(x \right)}}{\sin{\left(2 x \right)} + 3} = \frac{\sin{\left(x \right)}}{\sin{\left(2 x \right)} + 3} + \frac{\cos{\left(x \right)}}{\sin{\left(2 x \right)} + 3}$
2. Integrate term-by-term:
  1. Don't know the steps in finding this integral.
    
    But the integral is
    
    $\int \frac{\sin{\left(x \right)}}{\sin{\left(2 x \right)} + 3}\, dx$
  1. Don't know the steps in finding this integral.
    
    But the integral is
    
    $\int \frac{\cos{\left(x \right)}}{\sin{\left(2 x \right)} + 3}\, dx$
  The result is: $\int \frac{\sin{\left(x \right)}}{\sin{\left(2 x \right)} + 3}\, dx + \int \frac{\cos{\left(x \right)}}{\sin{\left(2 x \right)} + 3}\, dx$
Method #2
1. Rewrite the integrand:
  
  $\frac{\sin{\left(x \right)} + \cos{\left(x \right)}}{\sin{\left(2 x \right)} + 3} = \frac{\sin{\left(x \right)}}{2 \sin{\left(x \right)} \cos{\left(x \right)} + 3} + \frac{\cos{\left(x \right)}}{2 \sin{\left(x \right)} \cos{\left(x \right)} + 3}$
2. Integrate term-by-term:
  1. Don't know the steps in finding this integral.
    
    But the integral is
    
    $\frac{\sqrt{2} \left(\operatorname{atan}{\left(\frac{\sqrt{2} \tan{\left(\frac{x}{2} \right)}}{2} - \frac{\sqrt{2}}{2} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{4} - \frac{\sqrt{2} \left(\operatorname{atan}{\left(\frac{3 \sqrt{2} \tan{\left(\frac{x}{2} \right)}}{2} + \frac{\sqrt{2}}{2} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{4} - \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} - 2 \tan{\left(\frac{x}{2} \right)} + 3 \right)}}{8} + \frac{\log{\left(9 \tan^{2}{\left(\frac{x}{2} \right)} + 6 \tan{\left(\frac{x}{2} \right)} + 3 \right)}}{8}$
  1. Don't know the steps in finding this integral.
    
    But the integral is
    
    $- \frac{\sqrt{2} \left(\operatorname{atan}{\left(\frac{\sqrt{2} \tan{\left(\frac{x}{2} \right)}}{2} - \frac{\sqrt{2}}{2} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{4} + \frac{\sqrt{2} \left(\operatorname{atan}{\left(\frac{3 \sqrt{2} \tan{\left(\frac{x}{2} \right)}}{2} + \frac{\sqrt{2}}{2} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{4} - \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} - 2 \tan{\left(\frac{x}{2} \right)} + 3 \right)}}{8} + \frac{\log{\left(9 \tan^{2}{\left(\frac{x}{2} \right)} + 6 \tan{\left(\frac{x}{2} \right)} + 3 \right)}}{8}$
  The result is: $- \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} - 2 \tan{\left(\frac{x}{2} \right)} + 3 \right)}}{4} + \frac{\log{\left(9 \tan^{2}{\left(\frac{x}{2} \right)} + 6 \tan{\left(\frac{x}{2} \right)} + 3 \right)}}{4}$
Add the constant of integration:

$\int \frac{\sin{\left(x \right)}}{\sin{\left(2 x \right)} + 3}\, dx + \int \frac{\cos{\left(x \right)}}{\sin{\left(2 x \right)} + 3}\, dx+ \mathrm{constant}$

The answer is:

$\int \frac{\sin{\left(x \right)}}{\sin{\left(2 x \right)} + 3}\, dx + \int \frac{\cos{\left(x \right)}}{\sin{\left(2 x \right)} + 3}\, dx+ \mathrm{constant}$

The answer (Indefinite) [src]

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

$$\int {{{\sin x+\cos x}\over{\sin \left(2\,x\right)+3}}}{\;dx}$$

The answer [src]

  1                   
  /                   
 |                    
 |  cos(x) + sin(x)   
 |  --------------- dx
 |    3 + sin(2*x)    
 |                    
/                     
0

$$\int_{0}^{1}{{{\sin x+\cos x}\over{\sin \left(2\,x\right)+3}}\;dx}$$

  1                   
  /                   
 |                    
 |  cos(x) + sin(x)   
 |  --------------- dx
 |    3 + sin(2*x)    
 |                    
/                     
0

$$\int\limits_{0}^{1} \frac{\sin{\left(x \right)} + \cos{\left(x \right)}}{\sin{\left(2 x \right)} + 3}\, dx$$

Упростить

Numerical answer [src]

0.350522211844223

0.350522211844223

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

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Identical expressions

Similar expressions

Integral of (sinx+cosx)/(3+sin2x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2