Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of e^x/x^2
Integral of 1/√(4-x^2)
Integral of a^x*e^x
Integral of 1/(x^2+2*x+5)
Identical expressions
xe^xcos(8x)
xe to the power of x co sinus of e of (8x)
xexcos(8x)
xexcos8x
xe^xcos8x
xe^xcos(8x)dx

Solving integrals/ xe^xcos(8x)

Integral of xe^xcos(8x) dx

The solution

You have entered [src]

  1                 
  /                 
 |                  
 |     x            
 |  x*E *cos(8*x) dx
 |                  
/                   
0

$$\int\limits_{0}^{1} e^{x} x \cos{\left(8 x \right)}\, dx$$

Integral((x*E^x)*cos(8*x), (x, 0, 1))

Detail solution

Use integration by parts:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

Let $u{\left(x \right)} = x$ and let $\operatorname{dv}{\left(x \right)} = e^{x} \cos{\left(8 x \right)}$ .

Then $\operatorname{du}{\left(x \right)} = 1$ .

To find $v{\left(x \right)}$ :
1. Use integration by parts, noting that the integrand eventually repeats itself.
  1. For the integrand $e^{x} \cos{\left(8 x \right)}$ :
    
    Let $u{\left(x \right)} = \cos{\left(8 x \right)}$ and let $\operatorname{dv}{\left(x \right)} = e^{x}$ .
    
    Then $\int e^{x} \cos{\left(8 x \right)}\, dx = e^{x} \cos{\left(8 x \right)} - \int \left(- 8 e^{x} \sin{\left(8 x \right)}\right)\, dx$ .
  2. For the integrand $- 8 e^{x} \sin{\left(8 x \right)}$ :
    
    Let $u{\left(x \right)} = - 8 \sin{\left(8 x \right)}$ and let $\operatorname{dv}{\left(x \right)} = e^{x}$ .
    
    Then $\int e^{x} \cos{\left(8 x \right)}\, dx = 8 e^{x} \sin{\left(8 x \right)} + e^{x} \cos{\left(8 x \right)} + \int \left(- 64 e^{x} \cos{\left(8 x \right)}\right)\, dx$ .
  3. Notice that the integrand has repeated itself, so move it to one side:
    
    $65 \int e^{x} \cos{\left(8 x \right)}\, dx = 8 e^{x} \sin{\left(8 x \right)} + e^{x} \cos{\left(8 x \right)}$
    
    Therefore,
    
    $\int e^{x} \cos{\left(8 x \right)}\, dx = \frac{8 e^{x} \sin{\left(8 x \right)}}{65} + \frac{e^{x} \cos{\left(8 x \right)}}{65}$
Now evaluate the sub-integral.
Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{8 e^{x} \sin{\left(8 x \right)}}{65}\, dx = \frac{8 \int e^{x} \sin{\left(8 x \right)}\, dx}{65}$
  1. Use integration by parts, noting that the integrand eventually repeats itself.
    1. For the integrand $e^{x} \sin{\left(8 x \right)}$ :
      
      Let $u{\left(x \right)} = \sin{\left(8 x \right)}$ and let $\operatorname{dv}{\left(x \right)} = e^{x}$ .
      
      Then $\int e^{x} \sin{\left(8 x \right)}\, dx = e^{x} \sin{\left(8 x \right)} - \int 8 e^{x} \cos{\left(8 x \right)}\, dx$ .
    2. For the integrand $8 e^{x} \cos{\left(8 x \right)}$ :
      
      Let $u{\left(x \right)} = 8 \cos{\left(8 x \right)}$ and let $\operatorname{dv}{\left(x \right)} = e^{x}$ .
      
      Then $\int e^{x} \sin{\left(8 x \right)}\, dx = e^{x} \sin{\left(8 x \right)} - 8 e^{x} \cos{\left(8 x \right)} + \int \left(- 64 e^{x} \sin{\left(8 x \right)}\right)\, dx$ .
    3. Notice that the integrand has repeated itself, so move it to one side:
      
      $65 \int e^{x} \sin{\left(8 x \right)}\, dx = e^{x} \sin{\left(8 x \right)} - 8 e^{x} \cos{\left(8 x \right)}$
      
      Therefore,
      
      $\int e^{x} \sin{\left(8 x \right)}\, dx = \frac{e^{x} \sin{\left(8 x \right)}}{65} - \frac{8 e^{x} \cos{\left(8 x \right)}}{65}$
  So, the result is: $\frac{8 e^{x} \sin{\left(8 x \right)}}{4225} - \frac{64 e^{x} \cos{\left(8 x \right)}}{4225}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{e^{x} \cos{\left(8 x \right)}}{65}\, dx = \frac{\int e^{x} \cos{\left(8 x \right)}\, dx}{65}$
  1. Use integration by parts, noting that the integrand eventually repeats itself.
    1. For the integrand $e^{x} \cos{\left(8 x \right)}$ :
      
      Let $u{\left(x \right)} = \cos{\left(8 x \right)}$ and let $\operatorname{dv}{\left(x \right)} = e^{x}$ .
      
      Then $\int e^{x} \cos{\left(8 x \right)}\, dx = e^{x} \cos{\left(8 x \right)} - \int \left(- 8 e^{x} \sin{\left(8 x \right)}\right)\, dx$ .
    2. For the integrand $- 8 e^{x} \sin{\left(8 x \right)}$ :
      
      Let $u{\left(x \right)} = - 8 \sin{\left(8 x \right)}$ and let $\operatorname{dv}{\left(x \right)} = e^{x}$ .
      
      Then $\int e^{x} \cos{\left(8 x \right)}\, dx = 8 e^{x} \sin{\left(8 x \right)} + e^{x} \cos{\left(8 x \right)} + \int \left(- 64 e^{x} \cos{\left(8 x \right)}\right)\, dx$ .
    3. Notice that the integrand has repeated itself, so move it to one side:
      
      $65 \int e^{x} \cos{\left(8 x \right)}\, dx = 8 e^{x} \sin{\left(8 x \right)} + e^{x} \cos{\left(8 x \right)}$
      
      Therefore,
      
      $\int e^{x} \cos{\left(8 x \right)}\, dx = \frac{8 e^{x} \sin{\left(8 x \right)}}{65} + \frac{e^{x} \cos{\left(8 x \right)}}{65}$
  So, the result is: $\frac{8 e^{x} \sin{\left(8 x \right)}}{4225} + \frac{e^{x} \cos{\left(8 x \right)}}{4225}$
The result is: $\frac{16 e^{x} \sin{\left(8 x \right)}}{4225} - \frac{63 e^{x} \cos{\left(8 x \right)}}{4225}$
Now simplify:

$\frac{\left(65 x \left(8 \sin{\left(8 x \right)} + \cos{\left(8 x \right)}\right) - 16 \sin{\left(8 x \right)} + 63 \cos{\left(8 x \right)}\right) e^{x}}{4225}$
Add the constant of integration:

$\frac{\left(65 x \left(8 \sin{\left(8 x \right)} + \cos{\left(8 x \right)}\right) - 16 \sin{\left(8 x \right)} + 63 \cos{\left(8 x \right)}\right) e^{x}}{4225}+ \mathrm{constant}$

The answer is:

$\frac{\left(65 x \left(8 \sin{\left(8 x \right)} + \cos{\left(8 x \right)}\right) - 16 \sin{\left(8 x \right)} + 63 \cos{\left(8 x \right)}\right) e^{x}}{4225}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                                        
 |                          /          x      x         \       x                         x
 |    x                     |cos(8*x)*e    8*e *sin(8*x)|   16*e *sin(8*x)   63*cos(8*x)*e 
 | x*E *cos(8*x) dx = C + x*|----------- + -------------| - -------------- + --------------
 |                          \     65             65     /        4225             4225     
/

$$\int e^{x} x \cos{\left(8 x \right)}\, dx = C + x \left(\frac{8 e^{x} \sin{\left(8 x \right)}}{65} + \frac{e^{x} \cos{\left(8 x \right)}}{65}\right) - \frac{16 e^{x} \sin{\left(8 x \right)}}{4225} + \frac{63 e^{x} \cos{\left(8 x \right)}}{4225}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

   63    128*E*cos(8)   504*E*sin(8)
- ---- + ------------ + ------------
  4225       4225           4225

$$- \frac{63}{4225} + \frac{128 e \cos{\left(8 \right)}}{4225} + \frac{504 e \sin{\left(8 \right)}}{4225}$$

   63    128*E*cos(8)   504*E*sin(8)
- ---- + ------------ + ------------
  4225       4225           4225

$$- \frac{63}{4225} + \frac{128 e \cos{\left(8 \right)}}{4225} + \frac{504 e \sin{\left(8 \right)}}{4225}$$

-63/4225 + 128*E*cos(8)/4225 + 504*E*sin(8)/4225

Numerical answer [src]

0.293919384012818

0.293919384012818

The graph

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

Other calculators

How to use it?

Identical expressions

Integral of xe^xcos(8x) dx

Limits of integration:

The graph:

Piecewise:

The solution