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

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

The solution

You have entered [src]

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

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

Integral(cos(x)*exp(x), (x, 0, 1))

Detail solution

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

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

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

The answer is:

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

The answer (Indefinite) [src]

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

$$\int e^{x} \cos{\left(x \right)}\, dx = C + \frac{e^{x} \sin{\left(x \right)}}{2} + \frac{e^{x} \cos{\left(x \right)}}{2}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

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

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

-1/2 + E*cos(1)/2 + E*sin(1)/2

Numerical answer [src]

1.37802461354736

1.37802461354736

The graph

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

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

Limits of integration:

The graph:

Piecewise:

The solution