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Similar expressions
e^(x)*cos(x)*dx
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Solving integrals/ e^(-x)*cos(x)*dx

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

The solution

You have entered [src]

  1                
  /                
 |                 
 |   -x            
 |  e  *cos(x)*1 dx
 |                 
/                  
0

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

Simplify

Detail solution

Use integration by parts, noting that the integrand eventually repeats itself.
1. For the integrand $e^{- x} \cos{\left(x \right)} 1$ :
  
  Let $u{\left(x \right)} = \cos{\left(x \right)}$ and let $\operatorname{dv}{\left(x \right)} = e^{- x}$ .
  
  Then $\int e^{- x} \cos{\left(x \right)} 1\, dx = - \int e^{- x} \sin{\left(x \right)}\, dx - e^{- x} \cos{\left(x \right)}$ .
2. For the integrand $e^{- x} \sin{\left(x \right)}$ :
  
  Let $u{\left(x \right)} = \sin{\left(x \right)}$ and let $\operatorname{dv}{\left(x \right)} = e^{- x}$ .
  
  Then $\int e^{- x} \cos{\left(x \right)} 1\, dx = \int \left(- e^{- x} \cos{\left(x \right)}\right)\, dx + e^{- x} \sin{\left(x \right)} - e^{- x} \cos{\left(x \right)}$ .
3. Notice that the integrand has repeated itself, so move it to one side:
  
  $2 \int e^{- x} \cos{\left(x \right)} 1\, dx = e^{- x} \sin{\left(x \right)} - e^{- x} \cos{\left(x \right)}$
  
  Therefore,
  
  $\int e^{- x} \cos{\left(x \right)} 1\, dx = \frac{e^{- x} \sin{\left(x \right)}}{2} - \frac{e^{- x} \cos{\left(x \right)}}{2}$
Now simplify:

$- \frac{\sqrt{2} e^{- x} \cos{\left(x + \frac{\pi}{4} \right)}}{2}$
Add the constant of integration:

$- \frac{\sqrt{2} e^{- x} \cos{\left(x + \frac{\pi}{4} \right)}}{2}+ \mathrm{constant}$

The answer is:

$- \frac{\sqrt{2} e^{- x} \cos{\left(x + \frac{\pi}{4} \right)}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                             
 |                        -x                  -x
 |  -x                   e  *sin(x)   cos(x)*e  
 | e  *cos(x)*1 dx = C + ---------- - ----------
 |                           2            2     
/

$${{e^ {- x }\,\left(\sin x-\cos x\right)}\over{2}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

     -1                  -1
1   e  *sin(1)   cos(1)*e  
- + ---------- - ----------
2       2            2

$${{e^ {- 1 }\,\left(\sin 1-\cos 1\right)}\over{2}}+{{1}\over{2}}$$

     -1                  -1
1   e  *sin(1)   cos(1)*e  
- + ---------- - ----------
2       2            2

$$- \frac{\cos{\left(1 \right)}}{2 e} + \frac{\sin{\left(1 \right)}}{2 e} + \frac{1}{2}$$

Simplify

Numerical answer [src]

0.55539688265335

0.55539688265335

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(x)dx dx

Limits of integration:

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Similar expressions

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

Limits of integration:

The graph:

Piecewise:

The solution

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