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Integral of e^(-sin(x))*sin(2x)*dx dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1                     
  /                     
 |                      
 |   -sin(x)            
 |  E       *sin(2*x) dx
 |                      
/                       
0                       
$$\int\limits_{0}^{1} e^{- \sin{\left(x \right)}} \sin{\left(2 x \right)}\, dx$$
Integral(E^(-sin(x))*sin(2*x), (x, 0, 1))
Detail solution
  1. There are multiple ways to do this integral.

    Method #1

    1. The integral of a constant times a function is the constant times the integral of the function:

      1. Let .

        Then let and substitute :

        1. Use integration by parts:

          Let and let .

          Then .

          To find :

          1. The integral of the exponential function is itself.

          Now evaluate the sub-integral.

        2. The integral of the exponential function is itself.

        Now substitute back in:

      So, the result is:

    Method #2

    1. Rewrite the integrand:

    2. The integral of a constant times a function is the constant times the integral of the function:

      1. Let .

        Then let and substitute :

        1. Use integration by parts:

          Let and let .

          Then .

          To find :

          1. The integral of the exponential function is itself.

          Now evaluate the sub-integral.

        2. The integral of the exponential function is itself.

        Now substitute back in:

      So, the result is:

  2. Now simplify:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                         
 |                                                          
 |  -sin(x)                      -sin(x)      -sin(x)       
 | E       *sin(2*x) dx = C - 2*e        - 2*e       *sin(x)
 |                                                          
/                                                           
$$\int e^{- \sin{\left(x \right)}} \sin{\left(2 x \right)}\, dx = C - 2 e^{- \sin{\left(x \right)}} \sin{\left(x \right)} - 2 e^{- \sin{\left(x \right)}}$$
The graph
The answer [src]
       -sin(1)      -sin(1)       
2 - 2*e        - 2*e       *sin(1)
$$- \frac{2}{e^{\sin{\left(1 \right)}}} - \frac{2 \sin{\left(1 \right)}}{e^{\sin{\left(1 \right)}}} + 2$$
=
=
       -sin(1)      -sin(1)       
2 - 2*e        - 2*e       *sin(1)
$$- \frac{2}{e^{\sin{\left(1 \right)}}} - \frac{2 \sin{\left(1 \right)}}{e^{\sin{\left(1 \right)}}} + 2$$
2 - 2*exp(-sin(1)) - 2*exp(-sin(1))*sin(1)
Numerical answer [src]
0.412372289275322
0.412372289275322

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