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Solving integrals/ e^(-sin(x))*sin(x)cos(x)

Integral of e^(-sin(x))*sin(x)cos(x) dx

The solution

You have entered [src]

  1                          
  /                          
 |                           
 |   -sin(x)                 
 |  E       *sin(x)*cos(x) dx
 |                           
/                            
0

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

Integral((E^(-sin(x))*sin(x))*cos(x), (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. 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:
  
  $- e^{- \sin{\left(x \right)}} \sin{\left(x \right)} - e^{- \sin{\left(x \right)}}$
Method #2
1. Let $u = \sin{\left(x \right)}$ .
  
  Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
  
  $\int u e^{- u}\, du$
  1. Let $u = - u$ .
    
    Then let $du = - du$ 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 $v{\left(u \right)}$ :
      
      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:
    
    $- u e^{- u} - e^{- u}$
  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(x)                         -sin(x)    -sin(x)       
 | E       *sin(x)*cos(x) dx = C - e        - e       *sin(x)
 |                                                           
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

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

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

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

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

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

Numerical answer [src]

0.206186144637661

0.206186144637661

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

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Identical expressions

Similar expressions

Integral of e^(-sin(x))*sin(x)cos(x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2