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Similar expressions
sin(2x)/(2*exp(sinx))

Solving integrals/ sin(2x)/(2*exp(-sinx))

Integral of sin(2x)/(2*exp(-sinx)) dx

The solution

You have entered [src]

  1              
  /              
 |               
 |   sin(2*x)    
 |  ---------- dx
 |     -sin(x)   
 |  2*e          
 |               
/                
0

$$\int\limits_{0}^{1} \frac{\sin{\left(2 x \right)}}{2 e^{- \sin{\left(x \right)}}}\, dx$$

Integral(sin(2*x)/((2*exp(-sin(x)))), (x, 0, 1))

Detail solution

Rewrite the integrand:

$\frac{\sin{\left(2 x \right)}}{2 e^{- \sin{\left(x \right)}}} = e^{\sin{\left(x \right)}} \sin{\left(x \right)} \cos{\left(x \right)}$
Let $u = \sin{\left(x \right)}$ .

Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :

$\int u e^{u}\, du$
1. Use integration by parts:
  
  $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
  
  Let $u{\left(u \right)} = u$ and let $\operatorname{dv}{\left(u \right)} = e^{u}$ .
  
  Then $\operatorname{du}{\left(u \right)} = 1$ .
  
  To find $v{\left(u \right)}$ :
  1. The integral of the exponential function is itself.
    
    $\int e^{u}\, du = e^{u}$
  Now evaluate the sub-integral.
2. The integral of the exponential function is itself.
  
  $\int e^{u}\, du = e^{u}$
Now substitute $u$ back in:

$e^{\sin{\left(x \right)}} \sin{\left(x \right)} - e^{\sin{\left(x \right)}}$
Now simplify:

$\left(\sin{\left(x \right)} - 1\right) e^{\sin{\left(x \right)}}$
Add the constant of integration:

$\left(\sin{\left(x \right)} - 1\right) e^{\sin{\left(x \right)}}+ \mathrm{constant}$

The answer is:

$\left(\sin{\left(x \right)} - 1\right) e^{\sin{\left(x \right)}}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

$$\int \frac{\sin{\left(2 x \right)}}{2 e^{- \sin{\left(x \right)}}}\, dx = C + e^{\sin{\left(x \right)}} \sin{\left(x \right)} - e^{\sin{\left(x \right)}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

     sin(1)    sin(1)       
1 - e       + e      *sin(1)

$$- e^{\sin{\left(1 \right)}} + 1 + e^{\sin{\left(1 \right)}} \sin{\left(1 \right)}$$

     sin(1)    sin(1)       
1 - e       + e      *sin(1)

$$- e^{\sin{\left(1 \right)}} + 1 + e^{\sin{\left(1 \right)}} \sin{\left(1 \right)}$$

1 - exp(sin(1)) + exp(sin(1))*sin(1)

Numerical answer [src]

0.632248064512331

0.632248064512331

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

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Identical expressions

Similar expressions

Integral of sin(2x)/(2*exp(-sinx)) dx

Limits of integration:

The graph:

Piecewise:

The solution