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cos(x)*exp(x)
Solving integrals/ cos(x)*exp(x)

Integral of cos(x)*exp(x) dx

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The solution

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∫01excos⁡(x) dx\int\limits_{0}^{1} e^{x} \cos{\left(x \right)}\, dx 
Integral(cos(x)*exp(x), (x, 0, 1))
Detail solution

  1. Use integration by parts, noting that the integrand eventually repeats itself.

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

      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) dx=excos⁡(x)−∫(−exsin⁡(x)) dx\int e^{x} \cos{\left(x \right)}\, dx = e^{x} \cos{\left(x \right)} - \int \left(- e^{x} \sin{\left(x \right)}\right)\, dx.

    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) dx=exsin⁡(x)+excos⁡(x)+∫(−excos⁡(x)) dx\int e^{x} \cos{\left(x \right)}\, dx = e^{x} \sin{\left(x \right)} + e^{x} \cos{\left(x \right)} + \int \left(- e^{x} \cos{\left(x \right)}\right)\, dx.

    3. Notice that the integrand has repeated itself, so move it to one side:

      2∫excos⁡(x) dx=exsin⁡(x)+excos⁡(x)2 \int e^{x} \cos{\left(x \right)}\, dx = e^{x} \sin{\left(x \right)} + e^{x} \cos{\left(x \right)}

      Therefore,

      ∫excos⁡(x) dx=exsin⁡(x)2+excos⁡(x)2\int e^{x} \cos{\left(x \right)}\, dx = \frac{e^{x} \sin{\left(x \right)}}{2} + \frac{e^{x} \cos{\left(x \right)}}{2}

  2. Now simplify:

    2exsin⁡(x+π4)2\frac{\sqrt{2} e^{x} \sin{\left(x + \frac{\pi}{4} \right)}}{2}

  3. Add the constant of integration:

    2exsin⁡(x+π4)2+constant\frac{\sqrt{2} e^{x} \sin{\left(x + \frac{\pi}{4} \right)}}{2}+ \mathrm{constant}

The answer is:

2exsin⁡(x+π4)2+constant\frac{\sqrt{2} e^{x} \sin{\left(x + \frac{\pi}{4} \right)}}{2}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                        
 |                            x    x       
 |         x          cos(x)*e    e *sin(x)
 | cos(x)*e  dx = C + --------- + ---------
 |                        2           2    
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∫excos⁡(x) dx=C+exsin⁡(x)2+excos⁡(x)2\int e^{x} \cos{\left(x \right)}\, dx = C + \frac{e^{x} \sin{\left(x \right)}}{2} + \frac{e^{x} \cos{\left(x \right)}}{2}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.9002
Plot the graph f(x)
The answer [src] 
  1   E*cos(1)   E*sin(1)
- - + -------- + --------
  2      2          2
−12+ecos⁡(1)2+esin⁡(1)2- \frac{1}{2} + \frac{e \cos{\left(1 \right)}}{2} + \frac{e \sin{\left(1 \right)}}{2} 
=
= 
  1   E*cos(1)   E*sin(1)
- - + -------- + --------
  2      2          2
−12+ecos⁡(1)2+esin⁡(1)2- \frac{1}{2} + \frac{e \cos{\left(1 \right)}}{2} + \frac{e \sin{\left(1 \right)}}{2} 
-1/2 + E*cos(1)/2 + E*sin(1)/2
Numerical answer [src] 
1.37802461354736
1.37802461354736
The graph
Integral of cos(x)*exp(x) dx

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

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