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e^(-x)*cos(three *x)
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e(-x)*cos(3*x)
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Similar expressions
e^(x)*cos(3*x)

Solving integrals/ e^(-x)*cos(3*x)

Integral of e^(-x)cos(3x) dx

The solution

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  1                
  /                
 |                 
 |   -x            
 |  E  *cos(3*x) dx
 |                 
/                  
0

$$\int\limits_{0}^{1} e^{- x} \cos{\left(3 x \right)}\, dx$$

Integral(E^(-x)*cos(3*x), (x, 0, 1))

Detail solution

Let $u = - x$ .

Then let $du = - dx$ and substitute $- du$ :

$\int \left(- e^{u} \cos{\left(3 u \right)}\right)\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int e^{u} \cos{\left(3 u \right)}\, du = - \int e^{u} \cos{\left(3 u \right)}\, du$
  1. Use integration by parts, noting that the integrand eventually repeats itself.
    1. For the integrand $e^{u} \cos{\left(3 u \right)}$ :
      
      Let $u{\left(u \right)} = \cos{\left(3 u \right)}$ and let $\operatorname{dv}{\left(u \right)} = e^{u}$ .
      
      Then $\int e^{u} \cos{\left(3 u \right)}\, du = e^{u} \cos{\left(3 u \right)} - \int \left(- 3 e^{u} \sin{\left(3 u \right)}\right)\, du$ .
    2. For the integrand $- 3 e^{u} \sin{\left(3 u \right)}$ :
      
      Let $u{\left(u \right)} = - 3 \sin{\left(3 u \right)}$ and let $\operatorname{dv}{\left(u \right)} = e^{u}$ .
      
      Then $\int e^{u} \cos{\left(3 u \right)}\, du = 3 e^{u} \sin{\left(3 u \right)} + e^{u} \cos{\left(3 u \right)} + \int \left(- 9 e^{u} \cos{\left(3 u \right)}\right)\, du$ .
    3. Notice that the integrand has repeated itself, so move it to one side:
      
      $10 \int e^{u} \cos{\left(3 u \right)}\, du = 3 e^{u} \sin{\left(3 u \right)} + e^{u} \cos{\left(3 u \right)}$
      
      Therefore,
      
      $\int e^{u} \cos{\left(3 u \right)}\, du = \frac{3 e^{u} \sin{\left(3 u \right)}}{10} + \frac{e^{u} \cos{\left(3 u \right)}}{10}$
  So, the result is: $- \frac{3 e^{u} \sin{\left(3 u \right)}}{10} - \frac{e^{u} \cos{\left(3 u \right)}}{10}$
Now substitute $u$ back in:

$\frac{3 e^{- x} \sin{\left(3 x \right)}}{10} - \frac{e^{- x} \cos{\left(3 x \right)}}{10}$
Now simplify:

$\frac{\left(3 \sin{\left(3 x \right)} - \cos{\left(3 x \right)}\right) e^{- x}}{10}$
Add the constant of integration:

$\frac{\left(3 \sin{\left(3 x \right)} - \cos{\left(3 x \right)}\right) e^{- x}}{10}+ \mathrm{constant}$

The answer is:

$\frac{\left(3 \sin{\left(3 x \right)} - \cos{\left(3 x \right)}\right) e^{- x}}{10}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                   
 |                                 -x      -x         
 |  -x                   cos(3*x)*e     3*e  *sin(3*x)
 | E  *cos(3*x) dx = C - ------------ + --------------
 |                            10              10      
/

$$\int e^{- x} \cos{\left(3 x \right)}\, dx = C + \frac{3 e^{- x} \sin{\left(3 x \right)}}{10} - \frac{e^{- x} \cos{\left(3 x \right)}}{10}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

             -1      -1       
1    cos(3)*e     3*e  *sin(3)
-- - ---------- + ------------
10       10            10

$$\frac{3 \sin{\left(3 \right)}}{10 e} - \frac{\cos{\left(3 \right)}}{10 e} + \frac{1}{10}$$

             -1      -1       
1    cos(3)*e     3*e  *sin(3)
-- - ---------- + ------------
10       10            10

$$\frac{3 \sin{\left(3 \right)}}{10 e} - \frac{\cos{\left(3 \right)}}{10 e} + \frac{1}{10}$$

1/10 - cos(3)*exp(-1)/10 + 3*exp(-1)*sin(3)/10

Numerical answer [src]

0.151994333552281

0.151994333552281

The graph

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

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Identical expressions

Similar expressions

Integral of e^(-x)cos(3x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of e^(-x)*cos(3*x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Integral of e^(-x)cos(3x) dx