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

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

Limits of integration:

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Piecewise:

The solution

You have entered [src]
  1                
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 |  e  *cos(x)*1 dx
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01excos(x)1dx\int\limits_{0}^{1} e^{- x} \cos{\left(x \right)} 1\, dx
Detail solution
  1. Use integration by parts, noting that the integrand eventually repeats itself.

    1. For the integrand excos(x)1e^{- x} \cos{\left(x \right)} 1:

      Let u(x)=cos(x)u{\left(x \right)} = \cos{\left(x \right)} and let dv(x)=ex\operatorname{dv}{\left(x \right)} = e^{- x}.

      Then excos(x)1dx=exsin(x)dxexcos(x)\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 exsin(x)e^{- x} \sin{\left(x \right)}:

      Let u(x)=sin(x)u{\left(x \right)} = \sin{\left(x \right)} and let dv(x)=ex\operatorname{dv}{\left(x \right)} = e^{- x}.

      Then excos(x)1dx=(excos(x))dx+exsin(x)excos(x)\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:

      2excos(x)1dx=exsin(x)excos(x)2 \int e^{- x} \cos{\left(x \right)} 1\, dx = e^{- x} \sin{\left(x \right)} - e^{- x} \cos{\left(x \right)}

      Therefore,

      excos(x)1dx=exsin(x)2excos(x)2\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}

  2. Now simplify:

    2excos(x+π4)2- \frac{\sqrt{2} e^{- x} \cos{\left(x + \frac{\pi}{4} \right)}}{2}

  3. Add the constant of integration:

    2excos(x+π4)2+constant- \frac{\sqrt{2} e^{- x} \cos{\left(x + \frac{\pi}{4} \right)}}{2}+ \mathrm{constant}


The answer is:

2excos(x+π4)2+constant- \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     
/                                               
ex(sinxcosx)2{{e^ {- x }\,\left(\sin x-\cos x\right)}\over{2}}
The graph
0.001.000.100.200.300.400.500.600.700.800.902-2
The answer [src]
     -1                  -1
1   e  *sin(1)   cos(1)*e  
- + ---------- - ----------
2       2            2     
e1(sin1cos1)2+12{{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     
cos(1)2e+sin(1)2e+12- \frac{\cos{\left(1 \right)}}{2 e} + \frac{\sin{\left(1 \right)}}{2 e} + \frac{1}{2}
Numerical answer [src]
0.55539688265335
0.55539688265335
The graph
Integral of e^(-x)*cos(x)*dx dx

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